Thermodynamics and the Global Optimization of Lennard-Jones clusters

Theoretical design of global optimization algorithms can profitably utilize recent statistical mechanical treatments of potential energy surfaces (PES’s). Here we analyze the basin-hopping algorithm to explain its success in locating the global minima of Lennard-Jones (LJ) clusters, even those such as LJ38 for which the PES has a multiple-funnel topography, where trapping in local minima with different morphologies is expected. We find that a key factor in overcoming trapping is the transformation applied to the PES which broadens the thermodynamic transitions. The global minimum then has a significant probability of occupation at temperatures where the free energy barriers between funnels are surmountable.

[1]  Janet E. Jones,et al.  On the Calculation of Certain Crystal Potential Constants, and on the Cubic Crystal of Least Potential Energy , 1925 .

[2]  A. Mackay.  A dense non-crystallographic packing of equal spheres , 1962 .

[3]  I. R. Mcdonald,et al.  Calculation of thermodynamic properties of liquid argon from Lennard-Jones parameters by a Monte Carlo method , 1967 .

[4]  M. Hoare,et al.  Statistical mechanics and morphology of very small atomic clusters , 1976 .

[5]  G. Torrie,et al.  Nonphysical sampling distributions in Monte Carlo free-energy estimation: Umbrella sampling , 1977 .

[6]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[7]  M. Garey.  Johnson: computers and intractability: a guide to the theory of np- completeness (freeman , 1979 .

[8]  Thomas A. Weber,et al.  Hidden structure in liquids , 1982 .

[9]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[10]  L. Marks.  Surface structure and energetics of multiply twinned particles , 1984 .

[11]  Crystallization in curved three-dimensional space , 1984 .

[12]  L. Wille,et al.  Computational complexity of the ground-state determination of atomic clusters , 1985 .

[13]  Effect of topological frustration on the freezing temperature. , 1986, Physical review. B, Condensed matter.

[14]  J. Northby.  Structure and binding of Lennard‐Jones clusters: 13≤N≤147 , 1987 .

[15]  H. Scheraga,et al.  Monte Carlo-minimization approach to the multiple-minima problem in protein folding. , 1987, Proceedings of the National Academy of Sciences of the United States of America.

[16]  Alan M. Ferrenberg,et al.  New Monte Carlo technique for studying phase transitions. , 1988, Physical review letters.

[17]  F. Stillinger,et al.  Nonlinear optimization simplified by hypersurface deformation , 1988 .

[18]  Frank H. Stillinger,et al.  Supercooled liquids, glass transitions, and the Kauzmann paradox , 1988 .

[19]  Alan M. Ferrenberg,et al.  Optimized Monte Carlo data analysis. , 1989, Physical Review Letters.

[20]  D. Nelson,et al.  Polytetrahedral Order in Condensed Matter , 1989 .

[21]  D. L. Freeman,et al.  Reducing Quasi-Ergodic Behavior in Monte Carlo Simulations by J-Walking: Applications to Atomic Clusters , 1990 .

[22]  R. Whetten,et al.  Statistical thermodynamics of the cluster solid-liquid transition. , 1990, Physical review letters.

[23]  Frank H. Stillinger,et al.  Cluster optimization simplified by interaction modification , 1990 .

[24]  R. Zwanzig,et al.  Levinthal's paradox. , 1992, Proceedings of the National Academy of Sciences of the United States of America.

[25]  J. Onuchic,et al.  Protein folding funnels: a kinetic approach to the sequence-structure relationship. , 1992, Proceedings of the National Academy of Sciences of the United States of America.

[26]  Geometry, interaction range, and annealing , 1992 .

[27]  G. Parisi,et al.  Simulated tempering: a new Monte Carlo scheme , 1992, hep-lat/9205018.

[28]  C. J. Tsai,et al.  Use of an eigenmode method to locate the stationary points on the potential energy surfaces of selected argon and water clusters , 1993 .

[29]  J. Rose,et al.  KCl) 32 and the possibilities for glassy clusters , 1993 .

[30]  David J. Wales,et al.  Coexistence in small inert gas clusters , 1993 .

[31]  K. Re,et al.  Coexistence of multiple phases in finite systems. , 1993 .

[32]  Guoliang Xue.  Improvement on the northby algorithm for molecular conformation: Better solutions , 1994, J. Glob. Optim..

[33]  Thomas F. Coleman,et al.  A parallel build-up algorithm for global energy minimizations of molecular clusters using effective energy simulated annealing , 1993, J. Glob. Optim..

[34]  David J. Wales,et al.  Rearrangements of 55‐atom Lennard‐Jones and (C60)55 clusters , 1994 .

[35]  Berry,et al.  Multiple phase coexistence in finite systems. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[36]  F. Calvo,et al.  Configurational density of states from molecular dynamics simulations , 1995 .

[37]  Jonathan P. K. Doye,et al.  An order parameter approach to coexistence in atomic clusters , 1995 .

[38]  L. Piela,et al.  Molecular Dynamics on Deformed Potential Energy Hypersurfaces , 1995 .

[39]  Ho,et al.  Molecular geometry optimization with a genetic algorithm. , 1995, Physical review letters.

[40]  R. Zwanzig,et al.  Simple model of protein folding kinetics. , 1995, Proceedings of the National Academy of Sciences of the United States of America.

[41]  J. Doye,et al.  The effect of the range of the potential on the structures of clusters , 1995 .

[42]  F. Stillinger,et al.  A Topographic View of Supercooled Liquids and Glass Formation , 1995, Science.

[43]  J. Onuchic,et al.  Funnels, pathways, and the energy landscape of protein folding: A synthesis , 1994, Proteins.

[44]  Penna,et al.  Traveling salesman problem and Tsallis statistics. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[45]  Jonathan P. K. Doye,et al.  Calculation of thermodynamic properties of small Lennard‐Jones clusters incorporating anharmonicity , 1995 .

[46]  J. Doye,et al.  Magic numbers and growth sequences of small face-centered-cubic and decahedral clusters , 1995 .

[47]  C. Tsallis,et al.  Generalized simulated annealing , 1995, cond-mat/9501047.

[48]  Jonathan P. K. Doye,et al.  TOPICAL REVIEW: The effect of the range of the potential on the structure and stability of simple liquids: from clusters to bulk, from sodium to ? , 1996 .

[49]  Jonathan P. K. Doye,et al.  On potential energy surfaces and relaxation to the global minimum , 1996 .

[50]  Barkema,et al.  Event-Based Relaxation of Continuous Disordered Systems. , 1996, Physical review letters.

[51]  Howard R. Mayne,et al.  Global geometry optimization of atomic clusters using a modified genetic algorithm in space‐fixed coordinates , 1996 .

[52]  K. Ho,et al.  Structural optimization of Lennard-Jones clusters by a genetic algorithm , 1996 .

[53]  J. Onuchic,et al.  DIFFUSIVE DYNAMICS OF THE REACTION COORDINATE FOR PROTEIN FOLDING FUNNELS , 1996, cond-mat/9601091.

[54]  S. Gómez,et al.  Archimedean polyhedron structure yields a lower energy atomic cluster , 1996 .

[55]  Straub,et al.  Generalized simulated annealing algorithms using Tsallis statistics: Application to conformational optimization of a tetrapeptide. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[56]  Robert H. Leary,et al.  Global Optima of Lennard-Jones Clusters , 1997, J. Glob. Optim..

[57]  TOPOLOGY OF AMORPHOUS TETRAHEDRAL SEMICONDUCTORS ON INTERMEDIATE LENGTH SCALES , 1996, cond-mat/9610197.

[58]  I. Andricioaei,et al.  On Monte Carlo and molecular dynamics methods inspired by Tsallis statistics: Methodology, optimization, and application to atomic clusters , 1997 .

[59]  U. Hansmann.  Simulated annealing with Tsallis weights a numerical comparison , 1997, cond-mat/9710190.

[60]  J. Doye,et al.  Global Optimization by Basin-Hopping and the Lowest Energy Structures of Lennard-Jones Clusters Containing up to 110 Atoms , 1997, cond-mat/9803344.

[61]  Jonathan Doye,et al.  Global minima for transition metal clusters described by Sutton–Chen potentials , 1997 .

[62]  J. Doye,et al.  Surveying a potential energy surface by eigenvector-following , 1997 .

[63]  David Romero,et al.  Lower Energy Icosahedral Atomic Clusters with Incomplete Core , 1997 .

[64]  J. Doye,et al.  Structural consequences of the range of the interatomic potential A menagerie of clusters , 1997, cond-mat/9709201.

[65]  Jonathan Doye,et al.  Thermodynamics of Global Optimization , 1998 .

[66]  F. Calvo.  Chaos and dynamical coexistence in Lennard-Jones clusters , 1998 .

[67]  Howard R. Mayne,et al.  A study of genetic algorithm approaches to global geometry optimization of aromatic hydrocarbon microclusters , 1998 .

[68]  David J. Wales,et al.  Global minima of water clusters (H2O)n, n≤21, described by an empirical potential , 1998 .

[69]  Howard R. Mayne,et al.  An investigation of two approaches to basin hopping minimization for atomic and molecular clusters , 1998 .

[70]  Andrej Sali,et al.  Temperature dependence of the folding rate in a simple protein model: Search for a “glass” transition , 1998 .

[71]  G. Barkema,et al.  Traveling through potential energy landscapes of disordered materials: The activation-relaxation technique , 1997, cond-mat/9710023.

[72]  Mark A. Miller,et al.  Archetypal energy landscapes , 1998, Nature.