Topographies and Dynamics of Many-Dimensional Potential Surfaces

  • R. Stephen Berry
  • Ralph E. Kunz
Part of the NATO ASI Series book series (NSSE, volume 313)

Abstract

Multidimensional potential surfaces pose a variety of problems, not least of which is that it is now possible to obtain more information about the minima and other stationary points of such surfaces than we know how to use. This discussion describes a succession of steps to interpreting the topographies of such surfaces and of inferring the nature of the dynamics driven by those topographies. The procedure begins with a statistical search for minima and the saddles that link them. The topography lends itself to an analysis and categorization of local minima into large basins, including the primary basin, secondary and higher-order basins. From a representative sample of minima and saddles, one can construct inter-well rate coefficients and, from them, a master equation governing the flows on the surface. The flows can be categorized into intra-well and inter-well, and comparisons of the rates of these can be made, as the systems change temperature. A preliminary interpretation of the way the features of the topography govern the focusing or glassforming character of the surface.

Keywords

Global Minimum Potential Surface Master Equation Monotonic Sequence Distinct Minimum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Reference

  1. 1.
    Murrell, J.N. and Laidler, KJ. (1968) Trans. Farad. Soc. 64,371.CrossRefGoogle Scholar
  2. 2.
    Wales, DJ. and Berry, R.S. (1992) Limitations of the Murrell-Laidler theory, J. Chern. Soc. Farad. Trans. 8, 543–544.CrossRefGoogle Scholar
  3. 3.
    Stillinger, F.H. and Weber, T.A. (1982) Hidden Structure in Liquids, Phys. Rev. A 25, 978–989.CrossRefGoogle Scholar
  4. 4.
    Stillinger, F.H. and Weber, T.A. (1983) Dynamics of structural transitions in liquids, Phys. Rev. A 28, 2408–2416.CrossRefGoogle Scholar
  5. 5.
    Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vetterling, W.T. (1986) Numerical Recipes, Cambridge University Press, Cambridge.Google Scholar
  6. 6.
    Uppenbrink, J. and Wales, DJ. (1991) Packing Schemes for Lennard-Jones Clusters of 13 to 150 Atoms: Minima, Transition States and Rearrangement Mechanisms, J. Chern. Soc. Faraday Trans. 87, 215–222.CrossRefGoogle Scholar
  7. 7.
    Pancik, J. (1975) Calculation of the Least Energy Path on the Energy Hypersurface, ColI. Czech. Chern. Comm. 40, 1112–1118.Google Scholar
  8. 8.
    Hilderbrandt, R.L. (1977) Application of Newton-Raphson optimization techniques in molecular mechanics calculations, Cornput. Chern. 1, 179–186.Google Scholar
  9. 9.
    Cerjan, CJ. and Miller, W.H. (1981) On finding transition states, J. Chern. Phys. 75, 2800–2806.CrossRefGoogle Scholar
  10. 10.
    Simons, J., Jørgenson, P., Taylor, H. and Ozment, J. (1983) Walking on Potential Energy Surfaces, J. Phys. Chern. 87, 2745–2753.CrossRefGoogle Scholar
  11. 11.
    O’Neal, D., Taylor, H. and Simons, J. (1984) Potential Surface Walking and Reaction Paths for C2vBe + H2 <-BeH2 -= Be + 2H (1A1) J. Phys. Chern. 88, 1510–1513.CrossRefGoogle Scholar
  12. 12.
    Banerjee, A., Adams, N. and Simons, J. (1985) J. Phys. Chern. 89, 52.CrossRefGoogle Scholar
  13. 13.
    Baker, J. (1986) An Algorithm for the Location of Transition States, J. Comp. Chern. 7, 385–395.CrossRefGoogle Scholar
  14. 14.
    Baker, J. (1987) An Algorithm for Geometry Optimization Without Analytical Gradients, J. Comp. Chern. 8, 563–574.CrossRefGoogle Scholar
  15. 15.
    Shida, N., Barbara, P.F. and Almlof, J.E. (1989) J. Chern. Phys. 91, 4061.CrossRefGoogle Scholar
  16. 16.
    Metropolis, N., Metropolis, A.W., Rosenbluth, M.N., Teller, A.H. and Teller, E. (1953) J. Chern. Phys. 21, 1087.CrossRefGoogle Scholar
  17. 17.
    Saunders, M. (1989) Stochastic Search for the Conformations of Bicyclic Hydrocarbons,J. Comput. Chern. 10, 203–208.CrossRefGoogle Scholar
  18. 18.
    Saunders, M., Houk, K.N., Wu, Y.-D., Still, W.C., Lipton, M., Chang, G. and Guida, W. (1990) Conformations of Cycloheptadecane. A Comparison of Methods for Conformational Searching, J. Arn. Chern. Soc. 112, 1419–1427.CrossRefGoogle Scholar
  19. 19.
    Saunders, M. (1990) The Use of Stochastic Search in Looking for Homeomorphic Isomerism: Synthesis and Properties of Bicyclo[6.5.l]tetradecane, J. Arn. Chern. Soc. 112, 1791–1795.CrossRefGoogle Scholar
  20. 20.
    Stillinger, F.H. and Weber, T.A. (1984) Inherent pair correlation in simple liquids, J. Chern. Phys. 80, 4434–4437.CrossRefGoogle Scholar
  21. 21.
    McIver, J.W., Jr. and Komornicki, A. (1972) Structure of Transition States in Organic Reactions. General Theory and an Application to the Cyclobutene Butadiene Isomerization Using a Semiempirical Molecular Orbital Method, J. Arn. Chern. Soc. 94, 2625–2633.CrossRefGoogle Scholar
  22. 22.
    Komornicki, A. and McIver, J.W., Jr. (1973) J. Arn. Chern. Soc. 95, 4512.CrossRefGoogle Scholar
  23. 23.
    Komornicki, A. and McIver, J.W., Jr. (1974) Structure of Transition States. III. A MINDO/2 Study of the Cyclization of 1,3,5-Hexatriene to 1,3-Cyclohexadiene, J. Arn. Chern. Soc. 96, 5798–5800.CrossRefGoogle Scholar
  24. 24.
    McIver, J.W. (1974) The Structure of Transition States: Are They Symmetric?, Ace. Chern. Res. 7, 72–77.CrossRefGoogle Scholar
  25. 25.
    Poppinger, H. (1975) On the Calculation of Transition States, Chern. Phys. Lett. 35, 550–554.CrossRefGoogle Scholar
  26. 26.
    Halgren, T.A. and Lipscomb, W.N. (1977) The Synchronous Transit Method for Determining Reaction Pathways and Locating Molecular Transition States, Chern. Phys. Lett. 49, 225–232.CrossRefGoogle Scholar
  27. 27.
    Mezey, P.G., Peterson, M.R. and Csizmadia, I.G. (1977) Transition state determination by the X-method, Can. J. Chern. 55, 2941–2945.CrossRefGoogle Scholar
  28. 28.
    Murrell, J.N., Carter, S., Farantos, S.C., Huxley, P. and Varandas, A.I.C. (1984) Molecular Potential Energy Functions, Wiley, New York.Google Scholar
  29. 29.
    Li, S., Johnson, R.L. and Murrell, J.N. (1992) Cluster Structures and Stabilities from Solid-state Potentials, J. Chern. Soc., Faraday Trans. 88, 1229–1236.CrossRefGoogle Scholar
  30. 30.
    Maranas, C.D. and Floudas, C.A. (1992) A global optimization approach for Lennard-Jones microclusters, J. Chern. Phys. 97, 7667–7678.CrossRefGoogle Scholar
  31. 31.
    Wales, D.I. (1989) Finding saddle points for clusters, J. Chern. Phys. 91, 7002–7010.CrossRefGoogle Scholar
  32. 32.
    Wales, DJ. (1990) Transition States for Ar55, Chern. Phys. Lett. 166, 419–424.CrossRefGoogle Scholar
  33. 33.
    Berry, R.S., Davis, H.L. and Beck, T.L. (1988) Finding saddles on multidimensional potential surfaces, Chern. Phys. Lett. 147, 13–17.CrossRefGoogle Scholar
  34. 34.
    Davis, H.L., Wales, D.I. and Berry, R.S. (1990) Exploring potential energy surfaces with transition state calculations, J. Chern. Phys. 92, 4473–82.CrossRefGoogle Scholar
  35. 35.
    Hinde, R.I. and Berry, R.S. (1993) Chaotic dynamics and vibrational mode coupling in argon clusters, J. Chern. Phys. 99, 2942–2963.CrossRefGoogle Scholar
  36. 36.
    Hoare, M.R. and Pal, P. (1972) Nature 230, 5.Google Scholar
  37. 37.
    Hoare, M.R. and Pal, P. (1972) Nature 236, 75.CrossRefGoogle Scholar
  38. 38.
    Hoare, M.R. and Pal, P. (1972) J. Cryst. Growth 17, 77.CrossRefGoogle Scholar
  39. 39.
    Hoare, M.R. (1979) Structure and Dynamics of Simple Microclusters, Adv. Chern. Phys. 40, 49–135.CrossRefGoogle Scholar
  40. 40.
    Tsai, C.I. and Jordan, K.D. (1993) Use of the justogram and jump-walking methods for overcoming slow barrier crossing behavior in Monte Carlo simulations: Applications to the phase transitions in (Ar)13 and (H20)8 clusters, J. Chem. Phys. 99, 6957–6970.CrossRefGoogle Scholar
  41. 41.
    Tsai, CJ. and Jordan, K.D. (1993) Use of an Eigenmode Method To Locate the Stationary Points on the Potential Energy Surfaces of Selected Argon and Water Clusters, J. Phys. Chem. 97, 11227–11237.CrossRefGoogle Scholar
  42. 42.
    Frantz, D.D., Freeman, D.L. and Doll, J.D. (1990) Reducing quasi-ergodic behavior in Monte Carlo simulations by J-walking: Applications to atomic clusters, J. Chem. Phys. 93, 2769–2784.CrossRefGoogle Scholar
  43. 43.
    Frantz, D.D., Freeman, D.L. and Doll, J.D. (1992) Extending J-walking to quantum systems: application to atomic clusters, J. Chem. Phys. 97, 5713–5731.CrossRefGoogle Scholar
  44. 44.
    Braier, P.A., Berry, R.S. and Wales, DJ. (1990) How the range of pair potentials governs features of multidimensional potentials, J. Chem. Phys. 93, 8745–8756.CrossRefGoogle Scholar
  45. 45.
    Berry, R.S. and Breitengraser-Kunz, R.E. (1995) Topography and Dynamics of Multidimensional Interatomic Potential Surfaces, Phys. Rev. Lett. 74, 3951–3954.CrossRefGoogle Scholar
  46. 46.
    Kunz, R.E., Astakhova, T. and Berry, R.S. (1995) Using Clusters to Relate Topography and Dynamics of Multidimensional Potentials, Surf. Rev. Lett. (in press).Google Scholar
  47. 47.
    Kunz, R.E. and Berry, R.S. (1995) Statistical nterpretation of Topographies and Dynamics of Multidimensional Potentials, J. Chem. Phys. 103, 1904–1912.CrossRefGoogle Scholar
  48. 48.
    Doye, J.P.K., Wales, DJ. and Berry, R.S. (1995) The effect of the range of the potential on the Structures of Clusters, J. Chem. Phys. (in press).Google Scholar
  49. 49.
    Frauenfelder, H., Sligar, S.G. and Wolynes, P.G. (1991) The Energy Landscapes and Motions of Proteins, Science 254, 1598–1603.CrossRefGoogle Scholar
  50. 50.
    Bryngelson, J.D., Onuchic, J.N., Socci, N.D. and Wolynes, P.G. (1995) Funnels, Pathways and the Energy Landscape of Protein Folding, (submitted) Google Scholar
  51. 51.
    Rose, J.P. and Berry, R.S. (1993) Freezing, Melting, Nonwetting and Coexistence in (KCl)32 J. Chem. Phys. 98, 3246–3261.CrossRefGoogle Scholar
  52. 52.
    Rose, J.P. and Berry, R.S. (1993) (KCI)32 and the possibilities for glassy clusters, J. Chem. Phys. 98, 3262–3274.CrossRefGoogle Scholar
  53. 53.
    Cheng, H.-P. and Landman, U. (1993) Controlled Deposition, Soft Landing, and Glass Formation in Nanocluster-Surface Collisions, Science 260, 1304–1307.CrossRefGoogle Scholar
  54. 54.
    Amitrano, C. and Berry, R.S. (1992) Probability distributions of local Liapunov exponents for small clusters, Phys. Rev. Lett. 68, 729–732.CrossRefGoogle Scholar
  55. 55.
    Amitrano, C. and Berry, R.S. (1993) Probability distributions of local Liapunov exponents for Hamiltonian systems, Phys. Rev. E 47, 3158–3173.CrossRefGoogle Scholar
  56. 56.
    Beynon, J.H. and Gilbert, J.R. (1984) Application of Transition. State Theory to Unimolecular Reactions: An Introduction, Wiley, Chichester.Google Scholar
  57. 57.
    Robinson, P.J. and Holbrook, K.A. (1972) Unimolecular Reactions, WileyInterscience, London.Google Scholar
  58. 58.
    Jarrold, M.F. (1994) Introduction to Statistical Reaction Rate Theories, in H. Haberland (ed.), Clusters ofAtoms and Molecules, Springer-Verlag, Berlin.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • R. Stephen Berry
    • 1
  • Ralph E. Kunz
    • 2
  1. 1.The University of ChicagoChicagoUSA
  2. 2.Institut für Theoretische PhysikTechnische Universität BerlinBerlinGermany

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