A new and very efficient method for the large-scale global optimization of Lennard-Jones (L-J) clusters is presented in this paper. The problem, which consists of minimizing the potential energy of a cluster of particles that interact through the so-called L-J potential, is considered to be extremely difficult.

This new technique is original in that it combines the advantages of funnel-following methods, recently proposed by Leary, with a two-phase local search, described by Locatelli and Schoen [1]. In order to improve the probability of sampling good starting points, the local search tries to behave, in a sense, more globally, by using two penalty terms. The first term, acting on all pairs of atoms, is a spherical compression term, while the second term acts only on those few pairs of atoms whose distance is beyond a given threshold.

In this paper, the authors introduce a modification of the second term, allowing for greater flexibility; it is now possible to obtain an elliptical shape for the optimal L-J cluster. Numerical results demonstrate that, given a simple “standard” parameter setting, the new method is capable of locating most of the putative optima for N < 110, where N stands for the number of particles. The authors point out the ease with which the solutions to the cases N = 75, N = 98, and N = 102 are found, thanks to a suitable use of the elliptical penalty term. Surprisingly, those configurations, generally considered to be the most difficult ones, here become the easiest instances.