Abstract
We prove in this article lower bounds and upper bounds for the interatomic distance in cluster of atoms minimizing the Lennard-Jones energy. Our main result is in dimension three, but we also prove it in the two-dimensional case, since it seems interesting from a theoretical point of view.
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A. Ben Haj Yedder, X. Blanc, and C. Le Bris, “A numerical investigation of the 2-dimensional crystal problem,” Technical Report R03003, Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Jan. 2003. Available at http://www.ann.jussieu.fr/publications/2003.php3.
X. Blanc and C. Le Bris, “Crystalline order in one dimensional Thomas-Fermi type models,” in Mathematical Results in Quantum Mechanics (Taxco, 2001), vol. 307 of Contemp. Math., pp. 21–26. Amer. Math. Soc., Providence, RI, 2002.
X. Blanc and C. Le Bris, “Periodicity of the infinite-volume ground state of a one-dimensional quantum model,” Nonlinear Anal., Ser. A: Theory Methods, vol. 48, no. 6, pp. 791–803, 2002.
C.S. Gardner and C. Radin, “The infinite-volume ground state of the Lennard-Jones potential,” J. Statist. Phys., vol. 20, no. 6, pp. 719–724, 1979.
R.C. Heitmann and C. Radin, “The ground state for sticky disks,” J. Statist. Phys., vol. 22, no. 3, pp. 281–287, 1980.
J.E. Lennard-Jones and P.A. Taylor, “Some theoretical calculations of the physical properties of certain crystals,” Proc. Roy. Soc. London Series A, vol. 106, p. 476, 1925.
C. Radin, “The ground state for soft disks,” J. Statist. Phys., vol. 26, no. 2, pp. 365–373, 1981.
C. Radin, “Classical ground states in one dimension,” J. Statist. Phys., vol. 35, nos. 1/2, pp. 109–117, 1984.
C. Radin, “Low temperature and the origin of crystalline symmetry,” Internat. J. Modern Phys. B, vol. 1, nos. 5/6, pp. 1157–1191, 1987.
C. Radin and L.S. Schulman, “Periodicity of classical ground states,” Phys. Rev. Lett., vol. 51, no. 8, pp. 621–622, 1983.
D.J. Wales and H.A. Scheraga, “Global optimization of clusters, crystals, and biomolecules,” Science, vol. 285, pp. 1368–1372, 1999.
W.J. Ventevogel, “On the configuration of a one-dimensional system of interacting particles with minimum potential energy per particle,” Phys. A, vol. 92, nos. 3/4, pp. 343–361, 1978.
W.J. Ventevogel and B.R.A. Nijboer, “On the configuration of systems of interacting particles with minimum potential energy per particle,” Phys. A, vol. 99, no. 3, pp. 569–580, 1979.
W.J. Ventevogel and B.R.A. Nijboer, “On the configuration of systems of interacting particles with minimum potential energy per particle,” Phys. A, vol. 98, nos. 1/2, pp. 274–288, 1979.
G.L. Xue, “Minimum inter-particle distance at global minimizers of Lennard-Jones clusters,” J. Glob. Optim., vol. 11, no. 1, pp. 83–90, 1997.
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Blanc, X. Lower Bound for the Interatomic Distance in Lennard-Jones Clusters. Computational Optimization and Applications 29, 5–12 (2004). https://doi.org/10.1023/B:COAP.0000039486.97389.87
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DOI: https://doi.org/10.1023/B:COAP.0000039486.97389.87