Journal of Global Optimization

, Volume 39, Issue 4, pp 483–494 | Cite as

New bounds for Morse clusters

  • Tamás Vinkó
  • Arnold Neumaier
Open Access
Original Paper


This paper presents new, simple arguments improving the lower bounds for the total energy and the minimal inter-particle distance in minimal energy atom cluster problems with interactions given by a Morse potential, where the atom separation problem is difficult due to the finite energy at zero atom separation. Apart from being sharper than previously known bounds, they also apply for a wider range ρ ≥ 4.967 of the parameter in the Morse potential. Most results also hold for more general pair potentials.


Atom cluster Lower bounds Minimal distance Morse clusters 


  1. Blanc X. (2004). Lower bounds for the interatomic distance in Lennard–Jones clusters. Comput. Optim. Appl. 29: 5–12 CrossRefGoogle Scholar
  2. Bochner S. (1959) Lectures on Fourier Integrals. Princeton University Press, Princeton, NJGoogle Scholar
  3. Doye J.P.K., Leary R.H., Locatelli M., Schoen F. (2004). The global optimization of Morse clusters by potential energy transformations. INFORMS J. Comput. 16: 371–379 CrossRefGoogle Scholar
  4. Locatelli M., Schoen F. (2002). Minimal interatomic distance in Morse-clusters. J. Glob. Optim. 22: 175–190 CrossRefGoogle Scholar
  5. Maranas C., Floudas C. (1992). A global optimization approach for Lennard–Jones microclusters. J. Chem. Phys. 97: 7667–7678 CrossRefGoogle Scholar
  6. MuPAD Research Group. http://www.mupad.deGoogle Scholar
  7. Ruelle D. (1963). Classical statistical mechanics of a system of particles. Helv. Phys. Acta 36: 183–197 Google Scholar
  8. Ruelle D. (1969). Statistical Mechanics—Rigorous Results. Benjamin, New York Google Scholar
  9. Schachinger, W., Addis, B., Bomze, I.M., Schoen, F.: New results for molecular formation under pairwise potential minimization. Comput. Optim. Appl. to appear (in press)Google Scholar
  10. Vinkó, T.: Minimal inter-particle distance in atom clusters. Acta Cybern. 17, 105–119 (2005) Scholar
  11. Wolfram Research. Mathematica. Scholar
  12. Xue, G.L., Maier, R.S., Rosen, J.B.: Minimizing the Lennard–Jones potential function on a massively parallel computer. In: Proceedings of the 6th International Conference Supercomputing, Washington, DC, USA, pp. 409–416 (1992)Google Scholar
  13. Xue G.L. (1997). Minimum inter-particle distance at global minimizers of Lennard–Jones clusters. J. Glob. Optim. 11: 83–90 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Advanced Concepts TeamEuropean Space Agency, ESTECNoordwijkThe Netherlands
  2. 2.Fakultät für MathematikUniversität WienWienAustria

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