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Several organic compounds have resonance structures, but can a polymer have resonance structures?

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There's no reason why the structural subunits of polymers could not exhibit resonance, provided there are conjugated $\pi$-bonds present. A polymer such as kevlar, for example, certainly benefits from resonance stabilization within each monomeric unit.

Whether there exist any polymers in which the resonance is truly distributed over the entire molecule (if that's what you're really asking), however, I'm unsure. From the perspective of valence bond theory, the electron delocalization required for resonance necessitates a planar geometry with unhybridized $p$ orbitals that align and overlap to form $\pi$-bonds perpendicular to the plane of the $\sigma$-bond skeleton of the molecule. In molecular orbital theory, the sort of extensive electron delocalization that resonance implies requires a bonding molecular orbital, distributed over all participating atoms, constructed from a set of symmetrical, in-phase $p$ orbitals, with all planes of symmetry present in the original molecule(s) conserved in the generation of new molecular orbitals.

Given the rigidity of those criteria, I sincerely doubt that polymers of any great length could exist in which the resonance is distributed over the entire polymer chain. However, that could be just a failure of imagination or a lack of knowledge on my part, and I hope somebody else weighs in with a more conclusive answer.

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    $\begingroup$ In conjugated polymers, there is usually a finite conjugation length. That is, based on various properties, we know the delocalization is not infinite, even with fully planar, aligned chains like polyacetylene. Experimentally, vibrations and bond rotations make it unlikely to have very long aligned $\pi$ systems. Even in most electronic structure methods, there's a finite conjugation length, although it can be quite long. $\endgroup$ – Geoff Hutchison Sep 16 '14 at 18:49
  • $\begingroup$ @GeoffHutchison, thanks for your comment. At the time of my answer, my suspicion was that there were many practical reasons (variously physical, chemical, and thermodynamic) why the possible extent of $\pi$-conjugation in macromolecules is finite. On the other hand, I can't find any a priori line of reasoning as to why it must be limited (at least under some idealized set of conditions where some degrees of freedom are inaccessible and/or large energetic barriers to things like bond rotation exist). $\endgroup$ – Greg E. Sep 17 '14 at 16:46
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I rather like Greg E.'s answer. I'd like to add that there is at least one polymer that has resonances distributed over the entire chain: polyacetylene. When doped with iodine, its conductivity can reach values comparable to silver.

I feel like this answer might not have been what you were looking for, since I usually think of resonance forms as being distinct from each other, whereas for polyacetylene, the resonance forms look pretty much the same (just shifted over one bond).

https://en.wikipedia.org/wiki/Polyacetylene

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I feel it's important when considering this question to remember that resonance forms are a model, not an observable phenomenon -- they serve to plug the gap between observable electronic effects and Lewis structures.

Remembering that, it should be reasonably easy to see that multiple strengths and regions of conjugation, representable by more than one resonance form, could exist at various effective ratios and levels of robustness (according to chain deformation) in polymeric structures.

Consider a simple polyacetylene, for example: ${H_2C}\left({CH}\right)_n{CH_2}$. In the ideal case, this would have perfect conjugation along the entire straight chain, representable by two resonance forms. In practice, the electron density is asymmetric where the molecular geometry is also, which can be represented as additional resonance forms with charge separation.

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