Chemistry Reference
In-Depth Information
A transition will be characterized by a helical displacement of the electron if the
magnetic and electric transition dipoles are aligned. This is reflected in the Rosen-
feld equation for the CD intensity or
rotatory strength
,
R
a
→
j
, for a transition from a
ground state
a
to an excited state
j
in a collection of randomly-oriented molecules:
Im
R
a
→
j
=
a
|
μ
|
j
·
j
|
m
|
a
(6.121)
Straightforward application to the exciton bands yields:
R
1
A
1
→
1
A
2
=
√
2
πνρ
μ
2
R
1
A
1
→
1
E
=−
√
2
πνρ
μ
2
1
(6.122)
R
1
A
1
→
1
E
θ
=−
πνρ
μ
2
1
√
2
The out-of-plane polarized transition to the
1
A
2
state, which lies at higher energy,
has a positive CD signal, while the in-plane polarized transition to the lower
1
E
state
has a negative CD signal. The latter transition consists of two components along the
two in-plane directions. Summing over the three components in Eq. (
6.122
), shows
that the total rotatory strength, for randomly-oriented molecules, is exactly zero.
This is a general sum rule for CD spectra. If one now takes the spectrum of the chiral
antipode, the
Λ
tris-chelate complex, the spectra are exactly the same but the signs
are reversed. Mirror image in actual geometry thus becomes reflection symmetry in
the spectrum.
6.9 Induction Revisited: The Fibre Bundle
In Chap.
4
we left induction after the proof of the Frobenius reciprocity theorem.
In that proof the important concept of the positional representation was introduced.
This described the permutation of the sites under the action of the group elements.
Further, we defined local functions on the sites which transformed as irreps of the
site symmetry. As an example, if we want to describe the displacement of a cluster
atom in a polyhedron, two local functions are required: a totally-symmetric one for
the radial displacement and a twofold-degenerate one for the tangential displace-
ments. In cylindrical symmetry, these are labelled
σ
and
π
, respectively. The
me-
chanical
representation, i.e. the representation of the cluster displacements, is then
the sum of the two induced representations:
Γ
mech
=
Γ(σH
↑
G)
+
Γ(πH
↑
G)
(6.123)
As an example using the induction tables in Sect.
C.2
for an octahedron, we have:
Γ
mech
=
(A
1
g
+
E
g
+
T
1
u
)
+
(T
1
g
+
T
2
g
+
T
1
u
+
T
2
u
)
(6.124)
