On the Seven-particle Clusters of Lennard-Jones and Thomson
Abstract
In this Letter, we develop a graph-geometric framework and prove that the pentagonal bipyramid (PBP) is the unique global minimiser of both the seven-particle Lennard-Jones energy in ℝ3 and the seven-charge Thomson energy on S2, up to rigid symmetries and particle relabelling. For the Thomson problem, this strengthens a recent deterministic global-optimality certificate to numerical tolerance into an exact global-optimality and uniqueness theorem. A weak-interaction graph, defined by thresholding pair geometry, organises configuration space without imposing symmetry or discarding any pair interaction. For Lennard-Jones, pair-energy counting, graph classification, subset bounds, and force balance reduce all possible minimisers to 29 bounded domains; exact geometric certificates exclude 28, localise the survivor near the PBP, and a uniform restoring-energy estimate proves uniqueness. The Thomson proof follows the same progression: graph reduction, geometric localisation, and rigidity, but finalises through whole-domain coercivity and a global localisation theorem. The same fivefold order therefore emerges in two very different physical settings: an unconstrained cluster with a preferred bond length and purely repelling charges confined to a sphere, showing a common geometric organising principle for seven particles that is not tied to either interaction.
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BibTeX
@misc{shen2026sevenparticle,
title = {{On the Seven-particle Clusters of Lennard-Jones and Thomson}},
author = {Li Shen},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.09925},
eprint = {2611.09925},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.09925},
}
RIS
TY - UNPB
TI - On the Seven-particle Clusters of Lennard-Jones and Thomson
AU - Li Shen
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.09925
UR - https://arxiv.org/abs/2611.09925
AB - In this Letter, we develop a graph-geometric framework and prove that the pentagonal bipyramid (PBP) is the unique global minimiser of both the seven-particle Lennard-Jones energy in ℝ^3 and the seven-charge Thomson energy on S^2, up to rigid symmetries and particle relabelling. For the Thomson problem, this strengthens a recent deterministic global-optimality certificate to numerical tolerance into an exact global-optimality and uniqueness theorem. A weak-interaction graph, defined by thresholding pair geometry, organises configuration space without imposing symmetry or discarding any pair interaction. For Lennard-Jones, pair-energy counting, graph classification, subset bounds, and force balance reduce all possible minimisers to 29 bounded domains; exact geometric certificates exclude 28, localise the survivor near the PBP, and a uniform restoring-energy estimate proves uniqueness. The Thomson proof follows the same progression: graph reduction, geometric localisation, and rigidity, but finalises through whole-domain coercivity and a global localisation theorem. The same fivefold order therefore emerges in two very different physical settings: an unconstrained cluster with a preferred bond length and purely repelling charges confined to a sphere, showing a common geometric organising principle for seven particles that is not tied to either interaction.
ER -