Clar’s Rule Reveals the Topological Origin of Edge States in π-Conjugated Systems
Bibliographic record
Abstract
Edge states in π-conjugated systems govern scattering, magnetism, and fluorescence quenching, yet their origin has remained elusive. We show that Clar’s sextet-maximization principle, the resonance rule of aromatic hydrocarbons, captures the molecular and topological origins of edge states. Applied under boundary constraints, Clar’s rule predicts radicals at graphene edges and junctions in agreement with Z 2 topological invariants. Extending this framework to finite nanotubes, we derive a chirality-dependent counting rule and introduce explicit “Clar-cell” constructions that eliminate edge states. Excited-state calculations reveal that conventional unit-cell models host dark excitons from edge states, whereas Clar-cell nanotubes remain intrinsically bright even at nanometer lengths. These results resolve the paradox of ultrashort nanotube fluorescence and establish Clar’s rule as a molecular framework linking chemistry and topological physics of edge states.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".