2025/11/19 by Kiessling, Michael K. -H., Wales, David J.
Engineering · Mathematics · #Advanced Optimization Algorithms Research #Atomic and Molecular Clusters (physics.atm-clus) #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Spacecraft Dynamics and Control
paper · doi:10.48550/arxiv.2511.15008
openalex publication_date 2025/11/19 · openalex created_date 2025/11/23 · openalex updated_date 2026/07/28
Good a-priori bounds on the smallest pairwise distance r_\rmmin(LJN^\rmgmin) for a three-dimensional (3D) Lennard-Jones N-body cluster of globally minimal energy can significantly reduce the computational search space in the NP-hard problem to find this configuration. In this contribution the virial theorem is exploited for this purpose. We prove that if a configuration C(N) is a member of LJN^\rmequ (the stationary points), then r_\rmmin(C(N)) ≤ r_\rmmin(LJ2^\rmgmin). It is also shown that if C(N)∈ LJN^\rmgmin⊂ LJN^\rmequ, equality holds if and only if N∈\2,3,4\. We conjecture that r_\rmmin(LJN^\rmgmin) >1 in units for which r_\rmmin(LJ2^\rmgmin)= 2^\frac16 ≈ 1.122462048. This conjectured lower bound, if correct, would improve the best lower bound currently known, r_\rmmin(LJN^\rmgmin)≥ 0.767764, by about 25%. In these units the smallest minimal pair distance found through numerical searches for LJN^\rmgmin with N≤ 1000 is r_\rmmin(LJ923^\rmgmin) ≈ 1.01361, so the conjectured lower bound would presumably be close to optimal. From the virial theorem we obtain an identity for any C(N)∈ LJN^\rmequ, which expresses r_\rmmin(C(N)) in terms of the distribution of relative distances in C(N). This result reveals interesting connections with the Erdős distance, and related problems.