Forward and correct global optimization procedures for Lennard-Jones clusters

In this report some numerical results are presented for a special class of global optimization methods applied to molecular conformation problems. Following the results of a previously published method, in this report the authors show how some modifications to the basic structure of those algorithms enables us to obtain the optimal configuration for all clusters of Lennard-Jones atoms up to 80 atoms. 1 Problem definition The problem we are facing is that of determining the configuration of minimum potential energy of a cluster of identical atoms with no charge in threedimensional space. In this model all atoms are considered to be equal and only pairwise interaction is included in the definition of the potential energy. Let N ≥ 2 be an integer representing the total number of atoms. The Lennard-Jones (in short L-J) pairwise potential energy function is defined as follows: if the distance between the centers of a pair of atoms is r, then their contribution to the total energy is defined to be v(r) = 1 r12 − 2 r6 (1) and the L-J potential energy E of the cluster is defined as E(X) = E(X1, . . . , XN) = ∑ i<j v (‖Xi −Xj‖) (2) ∗Dip. di Informatica Universitá di Torino †Dip. di Sistemi e Informatica Universitá degli Studi di Firenze