Chemistry Reference
In-Depth Information
The central atom is invariant in
O
h
and thus transforms as
A
1
g
. The total induced
representation of the function space thus is given by
Γ
=
5
A
1
g
+
4
E
g
+
2
T
1
u
(4.153)
The 19-dimensional Hückel matrix thus will be resolved into five blocks, one of
dimension 5, two identical blocks of dimension 4, and three identical blocks of
dimension 2. In Table
4.9
we display the blocks for each irrep and the corresponding
SALCs for one component. The corresponding secular equations are:
λ
λ
4
13
2
8
λ
2
A
1
g
:
−
+
=
0
E
g
:
λ
4
5
λ
2
−
+
4
=
0
λ
2
T
1
u
:
−
1
=
0
(4.154)
Symmetry has taken us to a point where still quintic, quartic, and quadratic secular
equations must be solved. However, a closer look at this equations shows that they
can easily be solved. Apparently, a further symmetry principle is present, which
leads to simple analytical solutions of the secular equations. Triphenylmethyl is an
alternant
hydrocarbon. In an alternant, atoms can be given two different colors in
such a way that all bonds are between atoms of different colors; hence, no atoms
of the same color are adjacent. A graph with this property is
bipartite
,
5
and its
eigenvalue spectrum obeys the celebrated Coulson-Rushbrooke theorem [
16
].
Theorem 8
The eigenvalues of an alternant are symmetrically distributed about the
zero energy level
.
The corresponding eigenfunctions also show a mirror relation-
ship
,
except for a difference of sign
(
only
)
in every other atomic orbital coefficient
.
The total charge density at any carbon atom in the neutral alternant hydrocarbon
equals unity
.
Since triphenylmethyl is an odd alternant, there should be at least one eigenvalue
at energy zero. This root will be necessarily of
A
1
g
symmetry since this is the only
irrep that occurs an odd number of times. All other roots occur in pairs of opposite
energies. This is confirmed by the secular equations in Eq. (
4.154
), where the
A
1
g
equation indeed has a root at
λ
0, and all remaining equations contain only even
powers of
λ
. The roots are then easily determined (see also Fig.
4.10
):
=
4
√
3
A
1
g
:
λ
=
0
,
±
±
(4.155)
E
g
:
λ
=±
1
,
±
2
T
1
u
:
λ
=±
1
5
Note that in fact a molecular graph will always be bipartite unless it contains one or more odd-
membered rings.
