Chemistry Reference
In-Depth Information
D
z
z
C
2
=
p
z
|
C
2
|
p
z
3
p
x
|+
p
y
|+
p
z
|
C
2
|
p
x
+|
p
y
+|
p
z
1
=
3
p
x
|+
p
y
|+
p
z
|
|
p
x
−|
p
z
=
1
1
3
=
p
y
+|
(4.149)
An entirely similar derivation can be made for the
T
2
g
irrep, using the
d
-orbital set.
The results are:
E(T
1
u
)
=
β
(4.150)
E(T
2
g
)
=−
β
Triphenylmethyl Radical and Hidden Symmetry
As a final application, we discuss an example of a molecular radical, where more
symmetry is present than the eye meets. The triphenylmethyl radical, C
19
H
15
,isa
planar, conjugated, hydrocarbon-radical, with 19
π
-electrons. The molecular point
group for the planar configuration is
D
3
h
, but, since all valence 2
p
z
-orbitals are
antisymmetric with respect to the horizontal symmetry plane, the relevant symmetry
of the valence shell is only
C
3
v
as seen from Fig.
4.10
. The molecular symmetry
group distributes the 19 atoms over five trigonal orbits of atoms that, under
C
3
v
, can
solely be permuted with partners in the same orbit.
1. The central atom {o}.
2. The three atoms that are adjacent to
o
:
{
a,b,c
}
.
3. The six atoms in the
ortho
positions:
{
d,e,f,g,h,i
}
.
4. The six atoms in the
meta
positions:
{
p,q,r,s,t,u
}
.
5. The three atoms in the
para
positions:
.
The separate rotation of a single phenyl group by 180
◦
around its twofold direc-
tion will not change the connectivity of the graph. Yet this cannot be achieved by
elements of the point group. It is, however, a legitimate symmetry operation as far
as the graph is concerned since it preserves the connectivity. The resulting automor-
phism group is thus larger than the point group and in fact is isomorphic to
O
h
[
15
].
The three phenyl groups can be associated with the three Cartesian directions of this
octahedral group. The six atoms in the ortho orbit can be formally associated with
the six corners of this octahedron, each connected to a meta position (Fig.
4.10
). This
implies that the ortho and meta atoms occupy
C
4
v
sites. The three-atom orbits corre-
spond to the three tetragonal directions in the octahedron. The site group that leaves
such a tetragonal direction invariant is not of the conical
C
nv
type, but
D
4
h
. These
correspondences allow identification of all the permutations. As examples, the
C
4
symmetry element through the upper phenyl group and the
S
6
rotation-reflection
{
x,y,z
}
