1932

Abstract

In this review we discuss the recently described roaming mechanism for chemical reactions from the point of view of nonlinear dynamical systems in phase space. The recognition of the roaming phenomenon shows the need for further developments in our fundamental understanding of basic reaction dynamics, as is made clear by considering some questions that cut across most studies of roaming: Is the dynamics statistical? Can transition state theory be applied to estimate roaming reaction rates? What role do saddle points on the potential energy surface play in explaining the behavior of roaming trajectories? How do we construct a dividing surface that is appropriate for describing the transformation from reactants to products for roaming trajectories? How should we define the roaming region? We show that the phase space perspective on reaction dynamics provides the setting in which these questions can be properly framed and answered. We illustrate these ideas by considering photodissociation of formaldehyde. The phase-space formulation allows an unambiguous description of all possible reactive events, which also allows us to uncover the phase space mechanism that explains which trajectories roam, as opposed to evolving toward a different reactive event.

Loading

Article metrics loading...

/content/journals/10.1146/annurev-physchem-052516-050613
2017-05-05
2024-04-26
Loading full text...

Full text loading...

/deliver/fulltext/physchem/68/1/annurev-physchem-052516-050613.html?itemId=/content/journals/10.1146/annurev-physchem-052516-050613&mimeType=html&fmt=ahah

Literature Cited

  1. Bowman JM, Suits AG. 1.  2011. Roaming reactions: the third way. Phys. Today 64:33–37 [Google Scholar]
  2. Townsend D, Lahankar SA, Lee SK, Chambreau SD, Suits AG. 2.  et al. 2004. The roaming atom: straying from the reaction path in formaldehyde decomposition. Science 306:1158–61 [Google Scholar]
  3. van Zee RD, Foltz MF, Moore CB. 3.  1993. Evidence for a second molecular channel in the fragmentation of formaldehyde. J. Chem. Phys. 99:1664–73 [Google Scholar]
  4. Bowman JM, Shepler BC. 4.  2011. Roaming radicals. Annu. Rev. Phys. Chem. 62:531–53 [Google Scholar]
  5. Bowman JM. 5.  2014. Roaming. Mol. Phys. 112:2516–28 [Google Scholar]
  6. Suits AG. 6.  2008. Roaming atoms and radicals: a new mechanism in molecular dissociation. Acc. Chem. Res. 41:873–81 [Google Scholar]
  7. Herath N, Suits AG. 7.  2011. Roaming radical reactions. J. Phys. Chem. Lett. 2:642–47 [Google Scholar]
  8. Nakamura M, Tsai PY, Kasai T, Lin KC, Palazzetti F. 8.  et al. 2015. Dynamical, spectroscopic and computational imaging of bond breaking in photodissociation: roaming and role of conical intersections. Faraday Discuss. 177:77–98 [Google Scholar]
  9. Dhoke K, Zanni M, Harbola U, Venkatraman R, Arunan E. 9.  et al. 2015. Dynamics of chemical bond: general discussion. Faraday Discuss. 177:121–154 [Google Scholar]
  10. Waalkens H, Schubert R, Wiggins S. 10.  2008. Wigner's dynamical transition state theory in phase space: classical and quantum. Nonlinearity 21:R1 [Google Scholar]
  11. Mauguière FAL, Collins P, Kramer ZC, Carpenter BK, Ezra GS. 11.  et al. 2015. Phase space structures explain hydrogen atom roaming in formaldehyde decomposition. J. Phys. Chem. Lett. 6:4123–28 [Google Scholar]
  12. Houston PL, Conte R, Bowman JM. 12.  2016. Roaming under the microscope: trajectory study of formaldehyde dissociation. J. Phys. Chem. A 120:5103–14 [Google Scholar]
  13. Mauguière FAL, Collins P, Ezra GS, Farantos SC, Wiggins S. 13.  2014. Roaming dynamics in ketene isomerization. Theor. Chem. Acc. 133:1507–13 [Google Scholar]
  14. Mauguière FAL, Collins P, Kramer ZC, Carpenter BK, Ezra GS. 14.  et al. 2016. Phase space barriers and dividing surfaces in the absence of critical points of the potential energy: application to roaming in ozone. J. Chem. Phys. 144:054107 [Google Scholar]
  15. Mauguière FAL, Collins P, Stamatiadis S, Li A, Ezra GS. 15.  et al. 2016. Toward understanding the roaming mechanism in H + MgH → Mg + HH reaction. J. Phys. Chem. A 120:5145–54 [Google Scholar]
  16. Sun LP, Song KY, Hase WL. 16.  2002. A SN2 reaction that avoids its deep potential energy minimum. Science 296:875–78 [Google Scholar]
  17. Lopez JG, Vayner G, Lourderaj U, Addepalli SV, Kato S. 17.  et al. 2007. A direct dynamics trajectory study of F+CH3OOH reactive collisions reveals a major non-IRC reaction path. J. Am. Chem. Soc. 129:9976–85 [Google Scholar]
  18. Mikosch J, Trippel S, Eichhorn C, Otto R, Lourderaj U. 18.  et al. 2008. Imaging nucleophilic substitution dynamics. Science 319:183–86 [Google Scholar]
  19. Zhang J, Mikosch J, Trippel S, Otto R, Weidemüller M. 19.  et al. 2010. F + CH3I → FCH3 + I reaction dynamics. Nontraditional atomistic mechanisms and formation of a hydrogen-bonded complex. J. Phys. Chem. Lett. 1:2747–52 [Google Scholar]
  20. Rehbein J, Wulff B. 20.  2015. Chemistry in motion—off the MEP. Tetrahedron Lett. 177:6931–43 [Google Scholar]
  21. Klippenstein SJ, Georgievskii Y, Harding LB. 21.  2011. Statistical theory for the kinetics and dynamics of roaming reactions. J. Phys. Chem. A 115:14370–81 [Google Scholar]
  22. Audier HE, Morton TH. 22.  1993. Rearrangements in metastable ion decompositions of protonated propylamines. Org. Mass Spectrom. 28:1218–24 [Google Scholar]
  23. Heazlewood BR, Jordan MJT, Kable SH, Selby TM, Osborn DL. 23.  et al. 2008. Roaming is the dominant mechanism for molecular products in acetaldehyde photodissociation. PNAS 105:12719–24 [Google Scholar]
  24. Mezey PG. 24.  1987. Potential Energy Hypersurfaces Amsterdam: Elsevier
  25. Wales DJ. 25.  2003. Energy Landscapes Cambridge, UK: Cambridge University Press
  26. Pratihar S, Ma X, Homayoon Z, Barnes GL, Hase WL. 26.  2017. Direct chemical dynamics simulations. J. Am. Chem. Soc. 139:3570–90 [Google Scholar]
  27. Heidrich D. 27.  1995. The Reaction Path in Chemistry: Current Approaches and Perspectives New York: Springer
  28. Fukui K. 28.  1970. A formulation of the reaction coordinate. J. Phys. Chem. 74:4161–63 [Google Scholar]
  29. Pechukas P. 29.  1976. On simple saddle points of a potential surface, the conservation of nuclear symmetry along paths of steepest descent, and the symmetry of transition states. J. Chem. Phys. 64:1516–21 [Google Scholar]
  30. Evans MG, Polanyi M. 30.  1935. Some applications of the transition state method to the calculation of reaction velocities, especially in solution. Trans. Faraday Soc. 31:875–94 [Google Scholar]
  31. Eyring H. 31.  1935. The activated complex in chemical reactions. J. Chem. Phys. 3:107–15 [Google Scholar]
  32. Wigner EP. 32.  1938. The transition state method. Trans. Faraday Soc. 34:29–40 [Google Scholar]
  33. Rice OK, Ramsperger HC. 33.  1927. Theories of unimolecular gas reactions at low pressures. J. Am. Chem. Soc. 49:1617–29 [Google Scholar]
  34. Kassel LS. 34.  1932. The dynamics of unimolecular reactions. Chem. Rev. 10:11–25 [Google Scholar]
  35. Marcus RA. 35.  1952. Lifetimes of active molecules. I. J. Chem. Phys. 10:352–54 [Google Scholar]
  36. Marcus RA. 36.  1952. Lifetimes of active molecules. II. J. Chem. Phys. 10:355–59 [Google Scholar]
  37. Keck JC. 37.  1967. Variational theory of reaction rates. Adv. Chem. Phys. 13:85–121 [Google Scholar]
  38. Shepler BC, Braams BJ, Bowman JM. 38.  2008. Roaming dynamics in CH3CHO photodissociation revealed on a global potential energy surface. J. Phys. Chem. A 112:9344–51 [Google Scholar]
  39. Harding LB, Klippenstein SJ, Jasper AW. 39.  2012. Separability of tight and roaming pathways to molecular decomposition. J. Phys. Chem. A 116:6967–82 [Google Scholar]
  40. Robinson PJ, Holbrook KA. 40.  1972. Unimolecular Reactions Bristol, UK: Wiley-Interscience
  41. Steinfeld JI, Francisco JS, Hase WL. 41.  1989. Chemical Kinetics and Dynamics Englewood Cliffs, NJ: Prentice-Hall
  42. Gilbert RC, Smith SC. 42.  1990. Theory of Unimolecular and Recombination Reactions Oxford, UK: Blackwell Sci.
  43. Baer T, Hase WL. 43.  1996. Unimolecular Reaction Dynamics New York: Oxford Univ. Press
  44. Thiele E. 44.  1962. Comparison of the classical theories of unimolecular reactions. J. Chem. Phys. 36:1466–72 [Google Scholar]
  45. Pechukas P, Pollak E. 45.  1979. Classical transition state theory is exact if the transition state is unique. J. Chem. Phys. 71:2062–68 [Google Scholar]
  46. Miller WH. 46.  1976. Unified statistical model for “complex” and “direct” reaction mechanisms. J. Chem. Phys. 65:2216–23 [Google Scholar]
  47. Truhlar DG, Garrett BC. 47.  1980. Variational transition state theory. Acc. Chem. Res. 13:440–48 [Google Scholar]
  48. Pechukas P. 48.  1981. Transition state theory. Annu. Rev. Phys. Chem. 32:159–77 [Google Scholar]
  49. Truhlar DG, Garrett BC. 49.  1984. Variational transition state theory. Annu. Rev. Phys. Chem. 35:159–89 [Google Scholar]
  50. Hoffmann R, Swaminathan S, Odell BG, Gleiter R. 50.  1970. A potential surface for a nonconcerted reaction. Tetramethylene. J. Am. Chem. Soc. 92:7091–97 [Google Scholar]
  51. Doering WvE, Chen X, Lee K, Lin Z. 51.  2002. Fate of the intermediate diradicals in the caldera: stereochemistry of thermal stereomutations, (2 + 2) cycloreversions, and (2 + 4) ring-enlargements of cis- and trans-1-cyano-2-(E and Z)-propenyl-cis-3,4-dideuteriocyclobutanes. J. Am. Chem. Soc. 124:11642–52 [Google Scholar]
  52. Doering WvE, Barsa EW. 52.  2004. Fate of diradicals in the caldera: stereochemistry of thermal stereomutation and ring enlargement in cis- and trans-1-cyano-2(E)-propenylcyclopropanes. J. Am. Chem. Soc. 126:12353–62 [Google Scholar]
  53. Carpenter BK. 53.  2013. Energy disposition in reactive intermediates. Chem. Rev. 113:7265–86 [Google Scholar]
  54. Komatsuzaki T, Berry RS. 54.  2000. Local regularity and non-recrossing path in transition state: a new strategy in chemical reaction theories. J. Mol. Struct. 506:55–70 [Google Scholar]
  55. Wiggins S, Wiesenfeld L, Jaffé C, Uzer T. 55.  2001. Impenetrable barriers in phase-space. Phys. Rev. Lett. 86:5478–81 [Google Scholar]
  56. Uzer T, Jaffé C, Palacián J, Yanguas P, Wiggins S. 56.  2002. The geometry of reaction dynamics. Nonlinearity 15:957–92 [Google Scholar]
  57. Heidrich D, Quapp W. 57.  1986. Saddle points of index 2 on potential energy surfaces and their role in theoretical reactivity investigations. Theor. Chim. Acta 70:89–98 [Google Scholar]
  58. Ezra GS, Waalkens H, Wiggins S. 58.  2009. Microcanonical rates, gap times, and phase space dividing surfaces. J. Chem. Phys. 130:164118 [Google Scholar]
  59. Haller G, Palacián J, Yanguas P, Uzer T, Jaffé C. 59.  2010. Transition states near rank-two saddles: correlated electron dynamics of helium. Commun. Nonlinear Sci. Numer. Simul. 15:48–59 [Google Scholar]
  60. Shida N. 60.  2005. Onset dynamics of phase transition in Ar7. Adv. Chem. Phys. 130:129–53 [Google Scholar]
  61. Ezra G, Wiggins S. 61.  2009. Phase-space geometry and reaction dynamics near index 2 saddles. J. Phys. A 42:205101 [Google Scholar]
  62. Collins P, Ezra GS, Wiggins S. 62.  2011. Index k saddles and dividing surfaces in phase space with applications to isomerization dynamics. J. Chem. Phys. 134:244105 [Google Scholar]
  63. Mauguière F, Collins P, Ezra G, Wiggins S. 63.  2013. Bond breaking in a Morse chain under tension: fragmentation patterns, higher index saddles, and bond healing. J. Chem. Phys. 138:134118 [Google Scholar]
  64. Wiggins S. 64.  1988. Global Bifurcations and Chaos: Analytical Methods New York: Springer
  65. Wiggins S. 65.  1990. On the geometry of transport in phase space I. Transport in k degree-of-freedom Hamiltonian systems, 2≤k<∞. Physica D 44:471–501 [Google Scholar]
  66. Wiggins S. 66.  1994. Normally Hyperbolic Invariant Manifolds in Dynamical Systems New York: Springer
  67. Pechukas P, McLafferty FJ. 67.  1973. On transition state theory and the classical mechanics of collinear collisions. J. Chem. Phys. 58:1622–25 [Google Scholar]
  68. Pechukas P, Pollak E. 68.  1977. Trapped trajectories at the boundary of reactivity bands in molecular collisions. J. Chem. Phys. 67:5976–77 [Google Scholar]
  69. Pollak E, Pechukas P. 69.  1978. Transition states, trapped trajectories, and bound states embedded in the continuum. J. Chem. Phys. 69:1218–26 [Google Scholar]
  70. Waalkens H, Wiggins S. 70.  2004. Direct construction of a dividing surface of minimal flux for multi-degree-of-freedom systems that cannot be recrossed. J. Phys. A 37:L435 [Google Scholar]
  71. Wiggins S. 71.  1994. Normally Hyperbolic Invariant Manifolds in Dynamical Systems New York: Springer
  72. Meyer KR, Hall GR, Offin D. 72.  2009. Introduction to Hamiltonian Dynamical Systems and the N-Body Problem New York: Springer. 2nd ed.
  73. Li CB, Toda M, Komatsuzaki T. 73.  2009. Bifurcation of no-return transition states in many-body chemical reactions. J. Chem. Phys. 130:124116 [Google Scholar]
  74. Iñarrea M, Palacián JF, Pascual AI, Salas JP. 74.  2011. Bifurcations of dividing surfaces in chemical reactions. J. Chem. Phys. 135:014110 [Google Scholar]
  75. Mauguière FAL, Collins P, Ezra GS, Wiggins S. 75.  2013. Bifurcations of normally hyperbolic invariant manifolds in analytically tractable models and consequences for reaction dynamics. Int. J. Bifurcation Chaos 23:1330043 [Google Scholar]
  76. MacKay RS, Strub DC. 76.  2014. Bifurcations of transition states: Morse bifurcations. Nonlinearity 27:859–95 [Google Scholar]
  77. Bowman JM. 77.  2006. Skirting the transition state, a new paradigm in reaction rate theory. PNAS 103:16061–62 [Google Scholar]
  78. Wiesenfeld L, Faure A, Johann T. 78.  2003. Rotational transition states: relative equilibrium points in inelastic molecular collisions. J. Phys. B 36:1319–35 [Google Scholar]
  79. Wiesenfeld L. 79.  2004. Dynamics with a rotational transition state. Few-Body Syst. 34:163–68 [Google Scholar]
  80. Wiesenfeld L. 80.  2005. Geometry of phase space transition states: many dimensions, angular momentum. Adv. Chem. Phys. 130A:217–65 [Google Scholar]
  81. Chesnavich WJ, Bass L, Su T, Bowers MT. 81.  1981. Multiple transition states in unimolecular reactions: a transition state switching model. Application to the system. J. Chem. Phys. 74:2228–46 [Google Scholar]
  82. Chesnavich WJ, Bowers MT. 82.  1982. Theory of Ion-Neutral Interactions: Application of Transition State Theory Concepts to Both Collisional and Reactive Properties of Simple Systems Oxford, UK: Pergamon
  83. Chesnavich WJ. 83.  1986. Multiple transition states in unimolecular reactions. J. Chem. Phys. 84:2615–19 [Google Scholar]
  84. Langevin P. 84.  1905. A fundamental formula of kinetic theory. Ann. Chim. Phys. 5:245–88 [Google Scholar]
  85. Rynefors K, Markovic N. 85.  1985. Dynamics of centrifugal barrier complexes close to orbiting. Chem. Phys. 92:327–36 [Google Scholar]
  86. Child MS, Pfeiffer R, Baer M. 86.  1986. Ion-molecule collisions: nature and implications of trapped periodic orbits. Mol. Phys. 57:957–65 [Google Scholar]
  87. Hase WL, Wardlaw DL. 87.  1989. Transition state theory rate constants for association reactions without potential energy barriers. Bimolecular Collisons MNR Ashfold, JE Baggott 171–208 London: R. Soc. Chem. [Google Scholar]
  88. Mauguière FAL, Collins P, Ezra GS, Farantos SC, Wiggins S. 88.  2014. Multiple transition states and roaming in ion–molecule reactions: a phase space perspective. Chem. Phys. Lett. 592:282–87 [Google Scholar]
  89. Mauguière FAL, Collins P, Ezra GS, Farantos SC, Wiggins S. 89.  2014. Roaming dynamics in ion–molecule reactions: phase space reaction pathways and geometrical interpretation. J. Chem. Phys. 140:134112–17 [Google Scholar]
  90. Zhang X, Zou S, Harding LB, Bowman JM. 90.  2004. A global ab initio potential energy surface for formaldehyde. J. Phys. Chem. A 108:8980–86 [Google Scholar]
  91. Wiggins S. 91.  1991. Chaotic Transport in Dynamical Systems New York: Springer
  92. Davis M, Skodje R. 92.  1992. Chemical reactions as problems in nonlinear dynamics: review of statistical and adiabatic approximations from a phase space perspective. Intramolecular and Nonlinear Dynamics WL Hase 77–164 Stamford, CT: JAI
  93. Andrews DU, Kable SH, Jordan MJT. 93.  2013. A phase space theory for roaming reactions. J. Phys. Chem. A 117:7631–42 [Google Scholar]
  94. Lovejoy ER, Kim SK, Alvarez RA, Moore CB. 94.  1991. Kinetics of intramolecular carbon atom exchange in ketene. J. Chem. Phys. 95:4081–93 [Google Scholar]
  95. Lovejoy ER, Kim SK, Moore CB. 95.  1992. Observation of transition-state vibrational thresholds in the rate of dissociation of ketene. Science 256:1541–44 [Google Scholar]
  96. Lovejoy ER, Moore CB. 96.  1993. Structures in the energy dependence of the rate constant for ketene isomerization. J. Chem. Phys. 98:7846–54 [Google Scholar]
  97. Kirmse W. 97.  2002. 100 years of the Wolff rearrangement. Eur. J. Org. Chem. 2002:2193–256 [Google Scholar]
  98. Scott AP, Nobes RH, Schaefer HF III, Radom L. 98.  1994. The Wolff rearrangement: the relevant portion of the oxirene-ketene potential energy hypersurface. J. Am. Chem. Soc. 116:10159–64 [Google Scholar]
  99. Ulusoy IS, Stanton JF, Hernandez R. 99.  2013. Effects of roaming trajectories on the transition state theory rates of a reduced-dimensional model of ketene isomerization. J. Phys. Chem. A 117:7553–60 [Google Scholar]
  100. Ulusoy IS, Stanton JF, Hernandez R. 100.  2013. Correction to “Effects of roaming trajectories on the transition state theory rates of a reduced-dimensional model of ketene isomerization.”. J. Phys. Chem. A 117:10567–68 [Google Scholar]
  101. Gezelter JD, Miller WH. 101.  1995. Resonant features in the energy dependence of the rate of ketene isomerization. J. Chem. Phys. 103:7868–76 [Google Scholar]
  102. Ezra GS, Waalkens H, Wiggins S. 102.  2009. Microcanonical rates, gap times, and phase space dividing surfaces. J. Chem. Phys. 130:164118 [Google Scholar]
  103. Janssen C, Guenther J, Mauersberger K, Krankowsky D. 103.  2001. Kinetic origin of the ozone isotope effect: a critical analysis of enrichments and rate coefficients. Phys. Chem. Chem. Phys. 3:4718–21 [Google Scholar]
  104. Li H, Xiea D, Guo H. 104.  2004. An ab initio potential energy surface and vibrational states of MgH2 (11A′). J. Chem. Phys. 121:4156–63 [Google Scholar]
  105. Takayanagi T, Tanaka T. 105.  2011. Roaming dynamics in the MgH + H → Mg + H2 reaction: quantum dynamics calculations. Chem. Phys. Lett. 504:130–35 [Google Scholar]
  106. Malinova T, Guo ZX. 106.  2004. Artificial neural network modelling of hydrogen storage properties of Mg-based alloys. Mater. Sci. Eng. A 365:219–27 [Google Scholar]
  107. Song Y, Guo ZX, Yang R. 107.  2004. Influence of titanium on the hydrogen storage characteristics of magnesium hydride: a first principles investigation. Mater. Sci. Eng. A 365:73–79 [Google Scholar]
  108. Li A, Li J, Guo H. 108.  2013. Quantum manifestation of roaming in H + MgH → Mg + H2: the birth of roaming resonances. J. Phys. Chem. A 117:5052–60 [Google Scholar]
  109. Farantos SC, Schinke R, Guo H, Joyeux M. 109.  2009. Energy localization in molecules, bifurcation phenomena, and their spectroscopic signatures: the global view. Chem. Rev. 109:4248–71 [Google Scholar]
  110. Farantos SC. 110.  2014. Nonlinear Hamiltonian Mechanics Applied to Molecular Dynamics: Theory and Computational Methods for Understanding Molecular Spectroscopy and Chemical Reactions New York: Springer
  111. Harding LB, Klippenstein SJ, Jasper AW. 111.  2007. Ab initio methods for reactive potential surfaces. Phys. Chem. Chem. Phys. 9:4055–70 [Google Scholar]
  112. Shepler BC, Han Y, Bowman JM. 112.  2011. Are roaming and conventional saddle points for H2CO and CH3CHO dissociation to molecular products isolated from each other?. J. Phys. Chem. Lett. 2:834–38 [Google Scholar]
  113. Harding LB, Klippenstein SJ. 113.  2010. Roaming radical pathways for the decomposition of alkanes. J. Phys. Chem. Lett. 1:3016–20 [Google Scholar]
  114. Murrell JN, Laidler KJ. 114.  1968. Symmetries of activated complexes. Trans. Faraday Soc. 64:371–77 [Google Scholar]
  115. Wales DJ, Berry RS. 115.  1992. Limitations of the Murrell–Laidler theorem. J. Chem. Soc. Faraday Trans. 88:543–44 [Google Scholar]
  116. Nagahata Y, Teramoto H, Li C, Kawai S, Komatsuzaki T. 116.  2013. Reactivity boundaries to separate the fate of a chemical reaction associated with an index-two saddle. Phys. Rev. E 87:062817 [Google Scholar]
  117. Nagahata Y, Teramoto H, Li C, Kawai S, Komatsuzaki T. 117.  2013. Reactivity boundaries for chemical reactions associated with higher-index and multiple saddles. Phys. Rev. E 88:042923 [Google Scholar]
  118. Maronsson JB, Jónsson H, Vegge T. 118.  2012. A method for finding the ridge between saddle points applied to rare event rate estimates. Phys. Chem. Chem. Phys. 14:2884–91 [Google Scholar]
  119. Farantos SC. 119.  1998. POMULT: a program for computing periodic orbits in Hamiltonian systems based on multiple shooting algorithms. Comput. Phys. Commun. 108:240–58 [Google Scholar]
  120. Madrid JAJ, Mancho AM. 120.  2009. Distinguished trajectories in time dependent vector fields. Chaos 19:013111 [Google Scholar]
  121. Mendoza C, Mancho AM. 121.  2010. The hidden geometry of ocean flows. Phys. Rev. Lett. 105:038501 [Google Scholar]
  122. Mendoza C, Mancho AM. 122.  2012. The Lagrangian description of ocean flows: a case study of the Kuroshio current. Nonlin. Process. Geophys. 19:449–72 [Google Scholar]
  123. Mancho AM, Wiggins S, Curbelo J, Mendoza C. 123.  2013. Lagrangian descriptors: a method for revealing phase space structures of general time dependent dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 18:3530–57 [Google Scholar]
  124. Craven GT, Hernandez R. 124.  2015. Lagrangian descriptors of thermalized transition states on time-varying energy surfaces. Phys. Rev. Lett. 115:148301 [Google Scholar]
  125. Junginger A, Hernandez R. 125.  2015. Uncovering the geometry of barrierless reactions using Lagrangian descriptors. J. Phys. Chem. B 120:1720–25 [Google Scholar]
  126. Craven GT, Hernandez R. 126.  2016. Deconstructing field-induced ketene isomerization through Lagrangian descriptors. Phys. Chem. Chem. Phys. 18:4008–18 [Google Scholar]
/content/journals/10.1146/annurev-physchem-052516-050613
Loading
/content/journals/10.1146/annurev-physchem-052516-050613
Loading

Data & Media loading...

Supplemental Material

Supplementary Data

  • Article Type: Review Article
This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error