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The variational nature of the gentlest ascent dynamics and the relation of a variational minimum of a curve and the minimum energy path

  • Josep Maria BofillEmail author
  • Wolfgang Quapp
Regular Article

Abstract

It is shown that the path described by the gentlest ascent dynamics to find transition states (Weinan and Zhou in Nonlinearity 24:1831, 2011) is an example of a quickest nautical path for a given stationary wind or current, the so-called Zermelo navigation variational problem. In the present case, the current is the gradient of the potential energy surface. The result opens the possibility to propose new curves based on Zermelo’s theory for two tasks: locate transition states and define reaction paths. The relation between a minimal variational character, which some former reaction pathways possess, and the minimum energy path is discussed.

Keywords

Potential energy surface Reaction pathway Saddle point Steepest descent Newton trajectory Gradient extremal GAD 

References

  1. 1.
    Schlegel HB (2011) WIREs Comput Mol Sci 1:790CrossRefGoogle Scholar
  2. 2.
    Fukui K (1970) J Phys Chem 74:4161CrossRefGoogle Scholar
  3. 3.
    Quapp W, Heidrich D (1984) Theor Chim Acta 66:245CrossRefGoogle Scholar
  4. 4.
    Basilevsky MV, Shamov AG (1981) Chem Phys 60:347CrossRefGoogle Scholar
  5. 5.
    Hoffmann DK, Nord RS, Ruedenberg K (1986) Theor Chim Acta 69:265CrossRefGoogle Scholar
  6. 6.
    Quapp W (1989) Theor Chim Acta 75:447CrossRefGoogle Scholar
  7. 7.
    Schlegel HB (1992) Theor Chim Acta 83:15CrossRefGoogle Scholar
  8. 8.
    Rothman MJ, Lohr LL (1980) Chem Phys Lett 70:405CrossRefGoogle Scholar
  9. 9.
    Quapp W, Hirsch M, Imig O, Heidrich D (1998) J Comput Chem 19:1087CrossRefGoogle Scholar
  10. 10.
    Anglada JM, Besalú E, Bofill JM, Crehuet R (2001) J Comput Chem 22:387CrossRefGoogle Scholar
  11. 11.
    E W, Zhou X (2011) Nonlinearity 24:1831Google Scholar
  12. 12.
    Quapp W (2003) J Theor Comput Chem 2:385CrossRefGoogle Scholar
  13. 13.
    Crehuet R, Bofill JM (2005) J Chem Phys 122:234105CrossRefGoogle Scholar
  14. 14.
    Aguilar-Mogas A, Crehuet R, Giménez X, Bofill JM (2007) Mol Phys 105:2475CrossRefGoogle Scholar
  15. 15.
    Bofill JM, Quapp W, Caballero M (2012) J Chem Theory Comput 8:927CrossRefGoogle Scholar
  16. 16.
    Bofill JM, Quapp W (2011) J Chem Phys 134:074101CrossRefGoogle Scholar
  17. 17.
    Bofill JM, Quapp W, Caballero M (2013) Chem Phys Lett 583:203CrossRefGoogle Scholar
  18. 18.
    Courant R, Hilbert D (1953) Methods of mathematical physics. Interscience Publishers, New YorkGoogle Scholar
  19. 19.
    Quapp W (2003) Optimization 52:317CrossRefGoogle Scholar
  20. 20.
    Zermelo E (1931) Z Angew Math Mech 11:114CrossRefGoogle Scholar
  21. 21.
    Carathéodory C (1935) Variationsrechnung und partielle Differentialgleichungen erster Ordnung. B. G. Teubner, BerlinGoogle Scholar
  22. 22.
    Gibbons GW, Herdeiro CAR, Warnik CM, Werner MC (2009) Phys Rev D 79:044022CrossRefGoogle Scholar
  23. 23.
    Gibbons GW, Warnik CM (2011) Contemp Phys 52(3):197CrossRefGoogle Scholar
  24. 24.
    Randers G (1941) Phys Rev 59:195CrossRefGoogle Scholar
  25. 25.
    Pontryagin LS, Boltyanski VG, Gamkrelidze RV, Mishechenko EF (1962) The mathematical theory of optimal processes. Wiley, New YorkGoogle Scholar
  26. 26.
    Samanta A, Weinan E (2012) J Chem Phys 136:124104CrossRefGoogle Scholar
  27. 27.
    Crippen GM, Scheraga HA (1971) Arch Biochem Biophys 144:453CrossRefGoogle Scholar
  28. 28.
    Pancíř J (1975) Collect Czechoslov Chem Commun 40:1112CrossRefGoogle Scholar
  29. 29.
    Cerjan CJ, Miller WH (1981) J Chem Phys 75:2800CrossRefGoogle Scholar
  30. 30.
    Banerjee A, Adams N, Simons J, Shepard R (1985) J Phys Chem 89:52CrossRefGoogle Scholar
  31. 31.
    Baker J (1986) J Comput Chem 7:385CrossRefGoogle Scholar
  32. 32.
    Agrachev A, Gamkrelidze R (1998) Pure Appl Math 207:19Google Scholar
  33. 33.
    Bao D, Robles C, Shen Z (2004) J Differ Geom 66:377Google Scholar
  34. 34.
    Quapp W, Bofill JM, Bernuz E (2015) J Math Chem 53:41CrossRefGoogle Scholar
  35. 35.
    Neria E, Fischer S, Karplus M (1996) J Chem Phys 105:1902CrossRefGoogle Scholar
  36. 36.
    Hirsch M, Quapp W (2004) Chem Phys Lett 395:150CrossRefGoogle Scholar
  37. 37.
    Hratchian H, Frisch MJ, Schlegel HB (2010) J Chem Phys 133:224101CrossRefGoogle Scholar
  38. 38.
    Hirsch M, Quapp W (2004) J Math Chem 36:307CrossRefGoogle Scholar
  39. 39.
    Quapp W (1994) J Chem Soc Faraday Trans 90:1607Google Scholar
  40. 40.
    Quapp W (1996) Chem Phys Lett 253(3–4):286CrossRefGoogle Scholar
  41. 41.
    Branin FH (1972) IBM J Res Dev 16:504CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Departament de Química OrgànicaUniversitat de BarcelonaBarcelonaSpain
  2. 2.Institut de Química Teòrica i Computacional Universitat de Barcelona, (IQTCUB)BarcelonaSpain
  3. 3.Mathematisches InstitutUniversität LeipzigLeipzigGermany

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