Computational Optimization and Applications

, Volume 21, Issue 1, pp 55–70 | Cite as

Fast Global Optimization of Difficult Lennard-Jones Clusters

  • Marco Locatelli
  • Fabio Schoen

Abstract

The minimization of the potential energy function of Lennard-Jones atomic clusters has attracted much theoretical as well as computational research in recent years. One reason for this is the practical importance of discovering low energy configurations of clusters of atoms, in view of applications and extensions to molecular conformation research; another reason of the success of Lennard Jones minimization in the global optimization literature is the fact that this is an extremely easy-to-state problem, yet it poses enormous difficulties for any unbiased global optimization algorithm.

In this paper we propose a computational strategy which allowed us to rediscover most putative global optima known in the literature for clusters of up to 80 atoms and for other larger clusters, including the most difficult cluster conformations. The main feature of the proposed approach is the definition of a special purpose local optimization procedure aimed at enlarging the region of attraction of the best atomic configurations. This effect is attained by performing first an optimization of a modified potential function and using the resulting local optimum as a starting point for local optimization of the Lennard Jones potential.

Extensive numerical experimentation is presented and discussed, from which it can be immediately inferred that the approach presented in this paper is extremely efficient when applied to the most challenging cluster conformations. Some attempts have also been carried out on larger clusters, which resulted in the discovery of the difficult optimum for the 102 atom cluster and for the very recently discovered new putative optimum for the 98 atom cluster.

global optimization Lennard-Jones clusters molecular conformation 

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References

  1. 1.
    Anonymous, “The Leary tetrahedron: A new approach to global optimization,” NPACI and SDSC Online (http://www.npaci.edu/online/v3.19/leary.html), vol. 3, no. 19, pp. 1–4, 1999.Google Scholar
  2. 2.
    D. Deaven, N. Tit, J.R. Morris, and K.M. Ho, “Structural optimization of Lennard-Jones clusters by a genetic algorithm,” Chemical Physics Letters, vol. 256, no. 195, 1996.Google Scholar
  3. 3.
    J.P. Doye, M.A. Miller, and D.J. Wales, “The double funnel energy landscape of the 38-atom Lennard-Jones cluster,” The Journal of Chemical Physics, vol. 110, no. 14, pp. 6896–6906, 1999a.Google Scholar
  4. 4.
    J.P. Doye, D.J. Wales, and R.S. Berry, “The effect of the range of the potential on the structure of clusters,” Journal of Chemical Physics, vol. 103, no 10, pp. 4234–4249, 1995.Google Scholar
  5. 5.
    J.P.K. Doye, “The effect of compression on the global optimization of atomic clusters,” Physical Review E, vol. 62, pp. 8753–8761, 2000.Google Scholar
  6. 6.
    J.P.K. Doye, “The physics of global optimization of atomic clusters,” in Selected Case Studies in Global Optimization, J.D. Pinter (Ed.), Kluwer: Dordrecht, in press.Google Scholar
  7. 7.
    J.P.K. Doye, M.A. Miller, and D.J. Wales, “Evolution of the potential energy surface with size for the Lennard-Jones clusters,” Journal of Chemical Physics, vol. 111, pp. 8417–8428, 1999b.Google Scholar
  8. 8.
    C.A. Floudas and P.M. Pardalos, Handbook of Test Problems in Local and Global Optimization, vol. 33 of Nonconvex Optimization and its Applications. Kluwer Academic Publishers: Dordrecht.Google Scholar
  9. 9.
    J.C. Gilbert and J. Nocedal, “Global convergence properties of conjugate gradient methods,” SIAM Journal on Optimization, vol. 2, pp. 21–42, 1992.Google Scholar
  10. 10.
    M.S. Gockenbach, A.J. Kearsley, and W.W. Symes, “An infeasible point method for minimizing the Lennard-Jones potential,” Computational Optimization and Applications, vol. 8, pp. 273–286, 1997.Google Scholar
  11. 11.
    M. Hingst and A. Phillips, “A performance analysis of Appel's algorithm for performing pairwise calculations in a many particle system,” Computational Optimization and Applications, vol. 14, pp. 231–240, 1999.Google Scholar
  12. 12.
    M. Hoare, “Structure and dynamics of simple microclusters,” Advances in Chemical Physics, vol. 40, pp. 49–135, 1979.Google Scholar
  13. 13.
    R.H. Leary, “Global optima of Lennard-Jones clusters,” Journal of Global Optimization, vol. 11, no. 1, pp. 35–53, 1997.Google Scholar
  14. 14.
    R.H. Leary and J.P.K. Doye, “Tetrahedral global minimum for the 98 atom Lennard-Jones cluster,” Physical Review E, vol. 60, pp. R6320–R6322, 1999.Google Scholar
  15. 15.
    M. Locatelli and F. Schoen, “Forward and correct: global optimization procedures for Lennard-Jones clusters,” Technical Report 01/2001, Dipartimento di Sistemi e Informatica, Universitä di Firenze, 2001.Google Scholar
  16. 16.
    A. Neumaier, “Molecular modeling of proteins and mathematical prediction of protein structure,” SIAM Review, vol. 39, no. 3, pp. 407–460, 1997.Google Scholar
  17. 17.
    J.A. Northby, “Structure and binding of Lennard-Jones clusters: 13 ≤ n ≤ 147,” Journal of Chemical Physics, vol. 87, pp. 6166–6178, 1987.Google Scholar
  18. 18.
    J. Pillardy and L. Piela, “Molecular dynamics on deformed potential energy hypersurfaces,” Journal of Physical Chemistry, vol. 99, pp. 11805–11812, 1995.Google Scholar
  19. 19.
    W. Pullan, “Global optimization applied to molecular architecture,” PhD Thesis, Central Queensland University.Google Scholar
  20. 20.
    D.J. Wales and J.P.K. Doye, “Global optimization by Basin-Hopping and the lowest energy structures of Lennard-Jones clusters containing up to 110 atoms,” Journal of Physical Chemistry A, vol. 101, pp. 5111–5116, 1997.Google Scholar
  21. 21.
    D.J.Wales and H.A. Scheraga, “Global optimization of clusters, crystals, and biomolecules,” Science, vol. 285, no. 5432, pp. 1368–1372, 1999.Google Scholar
  22. 22.
    G. Xue, “Improvements on the Northby algorithm for molecular conformation: Better solutions,” Journal of Global Optimization, vol. 4, no. 4, pp. 425–440, 1994.Google Scholar
  23. 23.
    G.L. Xue, “Minimum inter-particle distance at global minimizers of Lennard-Jones clusters,” Journal of Global Optimization, vol. 11, no. 1, pp. 83–90, 1997.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Marco Locatelli
    • 1
  • Fabio Schoen
    • 2
  1. 1.Dipartimento di InformaticaUniversitá di TorinoTorinoItaly
  2. 2.Dipartimento di Sistemi e InformaticaUniversitá di FirenzeFirenzeItaly

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