Oligo(indenoindenes) (OInIn) are $\pi$-conjugated ladder carbon polymers with alternating hexagons and pentagons, hosting one unpaired electron for each of the latter in the open-shell limit. Here we study the main magnetic interactions in finite OInIn, classifying the six possible isomers in two different classes of three isomers each. We find that one class can be described by frustrated $S=1/2$ Heisenberg chains, with antiferromagnetic interactions between the second-neighbor sites. The other class is characterized by antiferromagnetic order. Employing several levels of theory, we further show that the ground state of one of the isomers is a valence-bond solid of ferromagnetic dimers ($S=1$). This is topologically similar to that of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, which is known to show fractional $S=1/2$ states at the edges.

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• Fractionalization
• Frustrated magnetism
• Magnetic interactions