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Proceedings of GO 2005, pp. 253 – 255.Improved lower bounds for optimization problems relatedto atom clusters ∗Tamás Vinkó 1 and Arnold Neumaier 21 Research Group on Artificial Intelligence of the Hungarian Academy of Sciences and University of Szeged, H-6720 Szeged,Aradi vértanúk tere 1., Hungary tvinko@inf.u-szeged.hu2 Institut für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria Arnold.Neumaier@univie.ac.atAbstractKeywords:A general method is introduced to obtain size independent lower bounds for the minimal interparticledistances in optimal atom cluster problems with pair potential functions. For the consideredpair potential function only reasonable properties are supposed. Derivation of explicit linear lowerbounds on the optimal energy values are also given. As demonstration new lower bounds for theLennard-Jones and Morse clusters are reported.Lower bounds, atom cluster problems, Lennard-Jones cluster, Morse cluster.1. IntroductionGiven a cluster of n atoms, define x i ∈ R d (i = 1, . . . , n, and d > 1) as the center of the ithatom. The potential energy of the cluster x = (x 1 , . . . , x n ) ∈ R dn is defined as the sum of thetwo-body inter-particle pair potentials over all of the pairs, i.e.,E(x) = ∑ i 2.r j = ‖x j − x 1 ‖ = ‖x j ‖2. Improved lower bounds(j = 1, . . . , n).We consider the particles in the cluster as balls and based on geometrial properties a lowerbound on the largest distance (measured between the origin and the balls) is given.∗ This work has been supported by the grants OTKA T 034350 and AÖU 56oü11.