New bounds and formulations for the deterministic global optimization of Lennard–Jones clusters

• Published in: Journal of global optimization
• Main Authors: Kuznetsov, Anatoliy, Sahinidis, Nikolaos V.
• Format: Journal Article
• Language: English
• Published: 11.03.2025
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-025-01476-7

What is the minimum-energy configuration of a cluster of identical atoms interacting with each other via the Lennard–Jones potential, a model of intermolecular forces between two charge-neutral species? Due to its fundamental importance in chemical physics, this question has been the subject of sustained study for over 50 years. A myriad of optimization algorithms have been developed to compute low-energy structures, i.e., local minima for this problem, and considerable efforts have gone towards characterizing the geometry of globally optimal solutions. Yet, the question of solving this problem to global optimality has remained elusive for all but the smallest clusters. In this work, we further refine the best known distance bounds on optimal Lennard–Jones clusters. We introduce a new formulation to eliminate symmetric solutions along with related strengthening inequalities and a convex underestimator of the Lennard–Jones potential. Together, these results enable the proof of global optimality for the putatively optimal 5- and 6-atom Lennard–Jones clusters by a general-purpose global optimization solver.