Chemistry Reference
In-Depth Information
The enantiomeric function reads:
x(t)
=
a
cos
nt
a
a
sin
nt
a
y(t)
=
−
z(t)
=
t
Note that a uniform sign change of
t
would leave the right-handed helix un-
changed. For the discrete helix, the screw symmetry consists of a translation in
the
z
-direction over a distance 2
πa/m
in combination with a rotation around
the
z
-axis over an angle 2
πn/m
.If
m
is irrational, the helix will not be periodic,
and the screw symmetry is lost.
4.1 The site symmetry of a cube is
T
h
. The cube is an invariant of its site group
and transforms as
a
g
in
T
h
. The set of five cubes thus spans the induced rep-
resentation:
aT
h
↑
I
h
. Applying the Frobenius theorem to the subduction (see
Sect.
C.1
), one obtains
aT
h
↑
I
h
=
A
g
+
G
g
(1)
4.2 The irreps can be obtained from the induction table in Sect.
C.2
,as
Γ
π
C
3
v
↑
T
d
:
Γ
π
C
3
v
↑
T
d
=
E
+
T
1
+
T
2
(2)
The SALCs shown span the tetrahedral
E
irrep, the one on the left is the
E
θ
component, and the one on the right is the
E
component. Note that they
transform into each other by rotating all
π
-orbitals over 90
◦
in the same sense
[
13
].
4.3 The 24 carbon atoms of coronene form three orbits: two orbits of six atoms,
corresponding to the internal hexagon and to the six atoms on the outer ring
that have bonds to the inner ring, and one orbit of the twelve remaining atoms.
The elements of the 6-orbit occupy sites of
C
2
v
symmetry, based on
C
2
,
σ
v
in
D
6
h
.The
p
z
orbitals on these sites transform as
b
1
, and hence the induced
irreps are as in the case of benzene:
σ
h
,
ˆ
ˆ
b
1
C
2
v
↑
D
6
h
=
B
2
g
+
A
2
u
+
E
1
g
+
E
2
u
(3)
The remaining 12-orbit connects carbon atoms with only
C
s
site symmetry,
the
p
z
orbitals on these sites transforming as
a
. The induced irreps read:
a
C
s
↑
D
6
h
=
B
1
g
+
B
2
g
+
A
1
u
+
A
2
u
+
2
E
1
g
+
2
E
2
u
(4)
The
A
1
u
and
B
1
g
irreps only appear in the 12-orbit, so we can infer that the
molecular orbitals with this symmetry will entirely be localized on the 12-
orbit. The SALCs can easily be constructed, as they should be antisymmetric
with respect to the
σ
v
planes in order not to hybridize with the SALCs based
ˆ
on the 6-orbits.
