A global optimization approach for Lennard-Jones microclusters

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Abstract
A global optimization approach is proposed for finding the global minimum energy configuration of Lennard-Jones microclusters. First, the original nonconvex total potential energy function, composed by rational polynomials, is transformed to the difference of two convex functions (DC transformation) via a novel procedure performed for each pair potential that constitute the total potential energy function. Then, a decomposition strategy based on the global optimization (GOP) algorithm [C. A. Floudas and V. Visweswaran, Comput. Chem. Eng. 14, 1397 (1990); V. Visweswaran and C. A. Floudas, ibid. 14, 1419 (1990); Proc. Process Systems Eng. 1991, I.6.1; C. A. Floudas and V. Visweswaran, J. Opt. Theory Appl. (in press)] is designed to provide tight bounds on the global minimum through the solutions of a sequence of relaxed dual subproblems. A number of theoretical results are included which expedite the computational effort by exploiting the special mathematical structure of the problem. The proposed approach attains ε convergence to the global minimum in a finite number of iterations. Based on this procedure, global optimum solutions are generated for small microclusters n≤7. For larger clusters 8≤N≤24 tight lower and upper bounds on the global solution are provided serving as excellent initial points for local optimization approaches. Finally, improved lower bounds on the minimum interparticle distance at the global minimum are provided.