Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Letter
  • Published:

Archetypal energy landscapes

Nature volume 394, pages 758–760 (1998)Cite this article

Abstract

Energy landscapes hold the key to understanding a wide range of molecular phenomena. The problem of how a denatured protein re-folds to its active state (Levinthal's parado1) has been addressed in terms of the underlying energy landscape2,3,4,5,6,7, as has the widely used ‘strong’ and ‘fragile’ classification of liquids8,9. Here we show how three archetypal energy landscapes for clusters of atoms or molecules can be characterized in terms of the disconnectivity graphs10 of their energy minima—that is, in terms of the pathways that connect minima at different threshold energies. First we consider a cluster of 38 Lennard–Jones particles, whose energy landscape is a ‘double funnel’ on which relaxation to the global minimum is diverted into a set of competing structures. Then we characterize the energy landscape associated with the annealing of C60 cages to buckministerfullerene, and show that it provides experimentally accessible clues to the relaxation pathway. Finally we show a very different landscape morphology, that of a model water cluster (H2O)20, and show how it exhibits features expected for a ‘strong’ liquid. These three examples do not exhaust the possibilities, and might constitute substructures of still more complex landscapes.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Figure 1: Pictorial correspondence between the potential-energy surface and the disconnectivity graph for three different energy landscapes, following Becker and Karplus10.
Figure 2: Disconnectivity graph for the double-funnel surface of (LJ)38.
Figure 3: The ‘pyracylene’ or ‘Stone–Wales’ rearrangement of C60.
Figure 4: Disconnectivity graph for minima and transition states in the five lowest Stone–Wales stacks of C60.
Figure 5: Disconnectivity graph for a sample of minima and transition states around the global minimum of the TIP4P (H2O)20 cluster.

Similar content being viewed by others

References

  1. Levinthal, C. in Mössbauer Spectroscopy in Biological Systems, Proceedings of a Meeting Held at Allerton House, Monticello, Illinois(eds DeBrunner, J. T. P. & Muck, E.) 22–24 (Univ. Illinois Press, Urbana, (1969)).

    Google Scholar 

  2. Ball, K. D. et al. From topography to dynamics on multidimensional potential energy surfaces of atomic clusters. Science 271, 963–966 (1996).

    Article  ADS  CAS  Google Scholar 

  3. Wales, D. J. Tales from topographic potential surfaces. Science 271, 925–929 (1996).

    Article  ADS  CAS  Google Scholar 

  4. Bryngelson, J. D., Onuchic, J. N., Socci, N. D. & Wolynes, P. G. Funnels, pathways, and the energy landscape of protein folding: a synthesis. Proteins 21, 167–195 (1995).

    Article  CAS  Google Scholar 

  5. Doye, J. P. K. & Wales, D. J. On potential energy surfaces and relaxation to the global minimum. J. Chem. Phys. 105, 8428–8445 (1996).

    Article  ADS  CAS  Google Scholar 

  6. Wolynes, P. G. Symmetry and the energy landscapes of biomolecules. Proc. Natl Acad. Sci. USA 93, 14249–14255 (1996).

    Article  ADS  CAS  Google Scholar 

  7. Berry, R. S., Elmaci, N., Rose, J. P. & Vekhter, B. Linking topography of its potential surface with the dynamics of folding of a protein model. Proc. Natl Acad. Sci. USA 94, 9520–9524 (1997).

    Article  ADS  CAS  Google Scholar 

  8. Angell, C. A. Formation of glasses from liquids and biopolymers. Science 267, 1924–1935 (1995).

    Article  ADS  CAS  Google Scholar 

  9. Stillinger, F. H. Atopographic view of supercooled liquids and glass formation. Science 267, 1935–1939 (1995).

    Article  ADS  CAS  Google Scholar 

  10. Becker, O. M. & Karplus, M. The topology of multidimensional potential energy surfaces: theory and application to peptide structure and kinetics. J. Chem. Phys. 106, 1495–1517 (1997).

    Article  ADS  CAS  Google Scholar 

  11. Wales, D. J. & Doye, J. P. K. Global optimization by basin-hopping and the lowest energy structuresof Lennard–Jones clusters containing up to 110 atoms. J. Phys. Chem. A 101, 5111–5116 (1997).

    Article  CAS  Google Scholar 

  12. Doye, J. P. K. & Wales, D. J. Surveying a potential energy surface by eigenvector-following. Z. Phys. D 40, 194–197 (1997).

    Article  ADS  CAS  Google Scholar 

  13. Kroto, H. W. Space, stars, C60, and soot. Science 242, 1139–1135 (1988).

    Article  ADS  CAS  Google Scholar 

  14. Heath, J. R., O'Brien, S. C., Curl, R. F., Kroto, H. W. & Smalley, R. E. Carbon condensation. Comments Cond. Mat. Phys. 13, 119–141 (1987).

    CAS  Google Scholar 

  15. Heath, J. R. in Fullerenes—Synthesis, Properties, and Chemistry of Large Carbon Clusters(eds Hammond, G. S. & Kuck, V. J.) 1–22 (ACS Symp. Ser. 481, Am. Chem. Soc., Washington DC, (1992)).

    Book  Google Scholar 

  16. Smalley, R. E. Self-assembly of the fullerenes. Acc. Chem. Res. 25, 98–105 (1992).

    Article  CAS  Google Scholar 

  17. Endo, M. & Kroto, H. W. Formation of carbon nanofibers. J. Phys. Chem. 96, 6941–6944 (1992).

    Article  CAS  Google Scholar 

  18. Murry, R. L., Strout, D. L., Odom, G. K. & Scuseria, G. E. Role of sp3 carbon and 7-membered rings in fullerene annealing and fragmentation. Nature 366, 665–667 (1993).

    Article  ADS  CAS  Google Scholar 

  19. Manopoulos, D. E. & Fowler, P. W. in The Far-reaching Impact of the Discovery of C60 (ed. Andreoni, W.) 51–69 (NATO ASI Ser. E 316, Kluwer, Dordrecht, (1993)).

    Google Scholar 

  20. Strout, D. L. & Scuseria, G. E. Acycloaddition model for fullerene formation. J. Phys. Chem. 100, 6492–6498 (1996).

    Article  CAS  Google Scholar 

  21. Marcos, P. A., López, M. J., Rubio, A. & Alonso, J. A. Thermal road for fullerene annealing. Chem. Phys. Lett. 273, 367–370 (1997).

    Article  ADS  CAS  Google Scholar 

  22. Ballone, P. & Milani, P. Simulated annealing of carbon clusters. Phys. Rev. B 42, 3201–3204 (1990).

    Article  ADS  CAS  Google Scholar 

  23. Chelikowsky, J. R. Nucleation of C60clusters. Phys. Rev. Lett. 67, 2970–2973 (1991).

    Article  ADS  CAS  Google Scholar 

  24. Chelikowsky, J. R. Formation of C60clusters via Langevin molecular-dynamics. Phys. Rev. B 45, 12062–12070 (1992).

    Article  ADS  CAS  Google Scholar 

  25. Jing, X. D. & Chelikowsky, J. R. Nucleation of carbon clusters via an accretion model. Phys. Rev. B 46, 5028–5031 (1992).

    Article  ADS  CAS  Google Scholar 

  26. Wang, C. Z., Xu, C. H., Chan, C. T. & Ho, K. M. Distintegration and formation of C60. J. Phys. Chem. 96, 3563–3565 (1992).

    Article  CAS  Google Scholar 

  27. Yi, J. Y. & Bernholc, J. Reactivity, stability, and formation of fullerenes. Phys. Rev. B 48, 5724–5727 (1993).

    Article  ADS  CAS  Google Scholar 

  28. Xu, C. H. & Scuseria, G. E. Tight-binding molecular-dynamics simulations of fullerene annealing and fragmentation. Phys. Rev. Lett. 72, 669–672 (1994).

    Article  ADS  CAS  Google Scholar 

  29. Maruyama, S. & Yamaguchi, Y. Amolecular dynamics demonstration of annealing to a perfect C60structure. Chem. Phys. Lett. 286, 343–349 (1998).

    Article  ADS  CAS  Google Scholar 

  30. Porezag, D., Frauenheim, T., Seifert, G. & Kaschner, R. Construction of tight-binding-like potentials on the basis of density-functional theory—application to carbon. Phys. Rev. B 51, 12947–12957 (1995).

    Article  ADS  CAS  Google Scholar 

  31. Austin, S. J., Fowler, P. W., Manolopoulos, D. E., Orlandi, G. & Zerbetto, F. Structural motifs and the stability of fullerenes. J. Phys. Chem. 99, 8076–8081 (1995).

    Article  CAS  Google Scholar 

  32. Austin, S. J., Fowler, P. W., Manolopoulos, D. E. & Zerbetto, F. The Stone–Wales map for C60. Chem. Phys. Lett. 235, 146–151 (1995).

    Article  ADS  CAS  Google Scholar 

  33. Stone, A. J. & Wales, D. J. Theoretical studies of icosahedral C60and some related species. Chem. Phys. Lett. 128, 501–503 (1986).

    Article  ADS  CAS  Google Scholar 

  34. Scuseria, G. E. Ab-initio calculations of fullerenes. Science 271, 942–945 (1996).

    Article  ADS  CAS  Google Scholar 

  35. Baker, J. & Fowler, P. W. Energetics of the Stone–Wales pyracylene transformation. J. Chem. Soc., Perkin Trans. 2, 1665–1666 (1992).

  36. Zhang, Q. L. et al. Reactivity of large carbon clusters—spheroidal carbon shells and their possible relevance to the formation and morphology of soot. J. Phys. Chem. 90, 525–528 (1986).

    Article  CAS  Google Scholar 

  37. Krätschmer, W., Lamb, L. D., Fostiropoulos, K. & Huffman, D. R. Solid C60—a new form of carbon. Nature 347, 354–358 (1990).

    Article  ADS  Google Scholar 

  38. Jorgensen, W. L. Quantum and statistical mechanical studies of liquids. 24. Revised TIPS for simulations of liquid water and aqueous-solutions. J. Chem. Phys. 77, 4156–4163 (1982).

    Article  ADS  CAS  Google Scholar 

  39. Wales, D. J. & Hodges, M. P. Global minima of water clusters (H2O)n, n ≤ 21, described by an empirical potential. Chem. Phys. Lett. 286, 65–72 (1998).

    Article  ADS  CAS  Google Scholar 

  40. Angell, C. A. Relaxation in liquids, polymers and plastic crystals—strong fragile patterns and problems. J. Non-Cryst. Solids 131–133;, 13–31 (1991).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We thank J. P. K. Doye and A. J. Stone for discussions. D.J.W., M.A.M. and T.R.W. thank the Royal Society, the EPSRC and the Cambridge Commonwealth Trust, respectively, for financial support.

Author information

Authors and Affiliations

  1. University Chemical Laboratories, Lensfield Road, Cambridge, CB2 1EW, UK

    David J. Wales, Mark A. Miller & Tiffany R. Walsh

Authors
  1. David J. Wales
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mark A. Miller
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tiffany R. Walsh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David J. Wales.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wales, D., Miller, M. & Walsh, T. Archetypal energy landscapes. Nature 394, 758–760 (1998). https://doi.org/10.1038/29487

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1038/29487

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing