Abstract
A stochastic global optimization method is applied to the challenging problem of finding the minimum energy conformation of a cluster of identical atoms interacting through the Lennard-Jones potential. The method proposed incorporates within an already existing and quite successful method, monotonic basin hopping, a two-phase local search procedure which is capable of significantly enlarging the basin of attraction of the global optimum. The experiments reported confirm the considerable advantages of this approach, in particular for all those cases which are considered in the literature as the most challenging ones, namely 75, 98, 102 atoms. While being capable of discovering all putative global optima in the range considered, the method proposed improves by more than two orders of magnitude the speed and the percentage of success in finding the global optima of clusters of 75, 98, 102 atoms.
Similar content being viewed by others
References
C. Barron, S. Gomez, and D. Romero, “Lower energy icosahedral atomic clusters with incomplete core,” Appl. Math. Lett., vol. 5, pp. 25-28, 1997.
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.
J.P.K. Doye, “The effect of compression on the global optimization of atomic clusters,” Physical Review E, vol. 62, pp. 8753-8761, 2000.
J.P.K. Doye, “Physical perspectives on the global optimization of atomic clusters,” in Selected Case Studies in Global Optimization. J.D. Pinter (Ed.), Dordrecht: Kluwer, 2002, p. in press.
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, 1999.
B. Hartke, “Efficient global geometry optimization of atomic and molecular clusters,” in Selected Case Studies in Global Optimization. J. Pinter (Ed.), Dordrecht: Kluwer Academic Publishers, 2002, p. in press.
R.H. Leary, “Global optima of Lennard-Jones clusters,” Journal of Global Optimization, vol. 11, no. 1, pp. 35-53, 1997.
R.H. Leary, “Global optimization on funneling landscapes,” Journal of Global Optimization, vol. 18, no. 4, pp. 367-383, 2000.
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.
D. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Mathematical Programming, vol. B 45, pp. 503-528, 1989.
M. Locatelli and F. Schoen, “Fast global optimization of difficult Lennard-Jones clusters,” Computational Optimization and Applications, vol. 21, no. 1, pp. 55-70, 2002.
J.J. More and Z. Wu, “Issues in large scale global molecular optimization,” in Large Scale Optimization with Applications: Part III: Molecular Structure and Optimization. L.T. Biegler, T.F. Coleman, A.R. Conn, and F.N. Santosa (Eds.), New York: Springer, 1997, pp. 99-121.
J.A. Northby, “Structure and binding of Lennard-Jones clusters: 13 ≤ n ≤ 147,” Journal of Chemical Physics, vol. 87, pp. 6166-6178, 1987.
J. Pillardy, A. Liwo, and H.A. Scheraga, “An efficient deformation-based global optimization method (selfconsistent basin-to-deformed-basin mapping (SCBDBM)). Application to Lennard-Jones atomic clusters,” J. Phys. Chem. A, vol. 103, pp. 9370-9377, 1999.
W. Pullan, “Global optimization applied to molecular architecture,” Ph.D. thesis, Central Queensland University, 1999.
C.S. Shao, R.H. Byrd, E. Eskow, and R.B. Schnabel, “Global optimization for molecular clusters using a new smoothing approach,” in Large Scale Optimization with Applications: Part III: Molecular Structure and Optimization. L.T. Biegler, T.F. Coleman, A.R. Conn, and F.N. Santosa (Eds.), New York: Springer, 1997, pp. 163-199.
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.
G.L. Xue, “Improvements on the Northby Algorithm for molecular conformation: Better solutions,” Journal of Global Optimization, vol. 4, no. 4, pp. 425-440, 1994.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Locatelli, M., Schoen, F. Efficient Algorithms for Large Scale Global Optimization: Lennard-Jones Clusters. Computational Optimization and Applications 26, 173–190 (2003). https://doi.org/10.1023/A:1025798414605
Issue Date:
DOI: https://doi.org/10.1023/A:1025798414605