98
4 Representations
Table 4.9 SALCs and Hückel matrices, in units of β, for triphenylmethyl. For the degenerate
irreps, only one component is given
A 1g
|o
1/
√
3(|a+|b+|c)
1/
√
3(|x+|y+|z)
1/
√
6(|d+|e+|f +|g+|h+|i)
1/
√
6(|p+|q+|r+|s+|t+|u)
−λ
√
3 000
√
3 −λ 0
√
20
00 −λ 0
√
2
0
√
20−λ 1
00
√
21−λ
E g
1/
√
2(|b−|c)
1/
√
2(|y−|z)
1/2(|f +|g−|h−|i)
1/2(|r+|s−|t−|u)
−λ 0
√
20
0 −λ 0
√
2
√
20−λ 1
0
√
21−λ
T 1u
1/
√
2(|d−|e)
1/
√
2(|p−|q)
−λ 1
1 −λ
The spectrum contains unexpected fivefold degeneracies at E =±β, where the T 1u
and E g levels coincide. This degeneracy is considered accidental, to the extent that
it does not correspond to a single irrep of the automorphism group of the graph.
However, the fivefold degeneracy can easily be rationalized as follows: E = β is the
eigenenergy of the degenerate highest occupied molecular orbital (HOMO) in an
isolated phenyl-ring. The three rings thus give rise to six orbitals with this energy.
The symmetry of this orbital space in O h is equal to A 1g + E g + T 1u . Of these only
the A 1g combination is of the right symmetry to interact with the central atom. For
the five others, there can be no communication between the phenyl rings since the
channel via the central atom is open only to A 1g symmetries. As a result, for these
solutions, there is no overlap between the rings, and five isolated phenyl solutions
persist at E = β. A similar argument applies to the level E =−β, which stems from
the phenyl lowest unoccupied orbital (LUMO).
Many features thus come together in triphenylmethyl. Besides the C 3v molecular
point group, the Hückel matrix obeys an additional or hidden O h symmetry. This
is a typical feature of the nearest-neighbor approximation, which requires only that
symmetry operations should preserve the connections with the nearest neighbors.
This precisely complies with the definition of the automorphism group of the graph.
Furthermore, the special bipartite properties of the graph further impose constraints
on the spectrum, which in this case lead to a complete reduction of the secular equations. Finally, an unexpected additional degeneracy manifests itself, which is related
4 Representations
Table 4.9 SALCs and Hückel matrices, in units of β, for triphenylmethyl. For the degenerate
irreps, only one component is given
A 1g
|o
1/
√
3(|a+|b+|c)
1/
√
3(|x+|y+|z)
1/
√
6(|d+|e+|f +|g+|h+|i)
1/
√
6(|p+|q+|r+|s+|t+|u)
−λ
√
3 000
√
3 −λ 0
√
20
00 −λ 0
√
2
0
√
20−λ 1
00
√
21−λ
E g
1/
√
2(|b−|c)
1/
√
2(|y−|z)
1/2(|f +|g−|h−|i)
1/2(|r+|s−|t−|u)
−λ 0
√
20
0 −λ 0
√
2
√
20−λ 1
0
√
21−λ
T 1u
1/
√
2(|d−|e)
1/
√
2(|p−|q)
−λ 1
1 −λ
The spectrum contains unexpected fivefold degeneracies at E =±β, where the T 1u
and E g levels coincide. This degeneracy is considered accidental, to the extent that
it does not correspond to a single irrep of the automorphism group of the graph.
However, the fivefold degeneracy can easily be rationalized as follows: E = β is the
eigenenergy of the degenerate highest occupied molecular orbital (HOMO) in an
isolated phenyl-ring. The three rings thus give rise to six orbitals with this energy.
The symmetry of this orbital space in O h is equal to A 1g + E g + T 1u . Of these only
the A 1g combination is of the right symmetry to interact with the central atom. For
the five others, there can be no communication between the phenyl rings since the
channel via the central atom is open only to A 1g symmetries. As a result, for these
solutions, there is no overlap between the rings, and five isolated phenyl solutions
persist at E = β. A similar argument applies to the level E =−β, which stems from
the phenyl lowest unoccupied orbital (LUMO).
Many features thus come together in triphenylmethyl. Besides the C 3v molecular
point group, the Hückel matrix obeys an additional or hidden O h symmetry. This
is a typical feature of the nearest-neighbor approximation, which requires only that
symmetry operations should preserve the connections with the nearest neighbors.
This precisely complies with the definition of the automorphism group of the graph.
Furthermore, the special bipartite properties of the graph further impose constraints
on the spectrum, which in this case lead to a complete reduction of the secular equations. Finally, an unexpected additional degeneracy manifests itself, which is related