Pierels instability

Infinite one dimensional conjugated pol3miers still maintain a finite bandgap due to a dimerization phenomonon known as the Pierels instability The free electron approximation (which views the pi electron region as an isotropic Drude electron gas) does not adequately explain this scenario. 

Fig. 2.6. The energy level diagram for (a) an unphysical infinite one dimensional metallic version of polyacetylene and (b) the more realistic dimerized polyacetylene molecule. The monomeric uniform molecule does not exhibit a bandgap because of its infinite size. A HOMO-LUMO gap exists despite the infinite extent of the molecule because of the Pierels instability which dimerizes the molecule and separates the bands at the Brilloin zone edge. These dispersion relationships are calculated using the derivation presented in [9] and [10].

Higher dimensional pi-electron carrier systems, like carbon nanotubes and graphene, are not subject to the Pierels instability and can be metallic or semi-metallic. 

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