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, C19H15, is a planar, conjugated, hydrocarbon-radical, with 19 tt-electrons. The molecular point group for the planar configuration is 3a, but, since all valence -orbitals are antisymmetric with respect to the horizontal symmetry plane, the relevant symmetry of the valence shell is only as seen from Fig. 4.10. The molecular symmetry group distributes the 19 atoms over five trigonal orbits of atoms that, under C3 , can solely be permuted with partners in the same orbit. 

A horizontal coordinate plane, dh, of the octahedron has the effect of flipping one single phenyl group around. We can at once determine the irreps of the different orbits by induction  

The central atom is invariant in Oh and thus transforms as Aig. The total induced representation of the function space thus is given by 

The 19-dimensional Hiickel 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  

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, and its eigenvalue spectrum obeys the celebrated Coulson-Rushbrooke theorem [16]. 

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