(10a) New Results in the Global Minimization of Molecular Potential Energy Functions

The problem of computing the global minimum of a molecular potential energy function has applications ranging from nanocluster physics [1] to biomolecule modeling [2], and has attracted considerable attention over the years [3-6] due to its challenging nature. This problem is highly nonconvex with many local minima and degrees of symmetry, and deterministic global optimization approaches have in the past been limited to very small systems [7]. Putative global minima have been found for a variety of systems [8], but the problem of certifying global optimality is still open.

In this work, we introduce novel convexification techniques for minimization problems involving the Lennard-Jones potential, which is commonly used to model van der Waals interactions. Additionally, we present optimality-based bounds on the inter-particle distances in Lennard-Jones clusters. In order to construct these bounds, we extend the argument presented by Blanc [9] and incorporate known upper bounds on the global minima. We present an open-source implementation of these results, and demonstrate their impact using the Lennard-Jones cluster structure prediction problem as a benchmark with the global optimization solver BARON. Finally, we discuss extensions of our results to other molecular potential energy functions.

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[2] Röder, K., Joseph, J.A., Husic, B.E. and Wales, D.J. Energy Landscapes for Proteins: From Single Funnels to Multifunctional Systems. Adv. Theory Simul., 2019, 2: 1800175.

[3] Hoare, M. R. Structure and Dynamics of Simple Microclusters. In Advances in Chemical Physics, Vol. 40; Rice, S. A., Ed.; Wiley, 1979; pp 49.

[4] Northby, J. A. (1987). Structure and Binding of Lennard-Jones Clusters: 13≤N≤147. J. Chem. Phys. 1987, 87, 6166-6177.

[5] Wales, D. J. and Doye, J. P. K. Global Optimization by Basin-Hopping and the Lowest Energy Structures of Lennard-Jones Clusters Containing up to 110 Atoms. J. Phys. Chem. A 1997, 101, 28, 5111-5116.

[6] Wang, Y., Lv, J., Zhu, L., Ma, Y. CALYPSO: A Method for Crystal Structure Prediction. Comput. Phys. Commun. 2012, 183(10), 2063-2070.

[7] Maranas, C. D. and Floudas, C. A. A Global Optimization Approach for Lennard-Jones Microclusters. J. Chem. Phys. 1992, 97, 7667-7678.

[8] Wales, D. J., Doye, J. P. K., Dullweber, A., Hodges, M. P., Naumkin, F. Y., Calvo, F., Hernández-Rojas, J., and Middleton, T. F. The Cambridge Energy Landscape Database. https://www-wales.ch.cam.ac.uk/CCD.html. Accessed Apr 6, 2022.

[9] Blanc, X. Lower Bound for the Interatomic Distance in Lennard-Jones Clusters. Comput. Optim. Appl. 2004, 29, 5-12.