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Variational Transition State Theory in Condensed Phases

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New Trends in Kramers’ Reaction Rate Theory

Part of the book series: Understanding Chemical Reactivity ((UCRE,volume 11))

Abstract

Variational transition state theory is a powerful tool for tackling the problem of activated barrier crossing in condensed phases, enabling the exact reduction of infinite-dimensional problems to much more tractable few-dimensional problems. Variational transition state theory may be used to go beyond Kramers’ theory and consider the effects of memory friction and solute nonlinearities on the rate of barrier crossing. In addition, since variational transition state theory is a Hamiltonian based formalism, it is not tied to the limitations of generalized Langevin equation dynamics; thus it can be used to treat nonlinear solute-solvent interactions. Other recent advances include the development of new formalisms for improved optimization of the transition state dividing surface, the incorporation of intramolecular solute modes, and the development of centroid-based quantum transition state theory analogs.

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References

  1. H.A. Kramers, Physica 7, 284 (1940).

    Article  CAS  Google Scholar 

  2. P. Hänggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 250 (1990).

    Article  Google Scholar 

  3. E. Pollak, in Activated Barrier Crossing, P. Hänggi and G. Fleming (Eds.), World Scientific, Singapore (1992).

    Google Scholar 

  4. B. J. Berne, M. Borkovec, and J. E. Straub, J. Phys. Chem. 92, 3711 (1988).

    Article  CAS  Google Scholar 

  5. J. T. Hynes, in Theory of Chemical Reaction Dynamics, M. Baer (Ed.), p. 171, CRC Press, Boca Raton, FL (1985).

    Google Scholar 

  6. A. Nitzan, J. Chem. Phys. 82, 1614 (1985).

    Article  CAS  Google Scholar 

  7. D. G. Truhlar, W. L. Hase, and J. T. Hynes, J. Phys. Chem 87, 2664, 5523(E) (1983).

    Article  CAS  Google Scholar 

  8. R. F. Grote and J. T. Hynes, J. Chem. Phys. 73, 2715 (1980).

    Article  CAS  Google Scholar 

  9. N. Wax, Selected Papers on Noise and Stochastic Processes, Dover, New York (1954).

    Google Scholar 

  10. H. Risken, The Fokker-Planck Equation, 2nd ed., Springer-Verlag, Berlin (1989).

    Book  Google Scholar 

  11. R. Serra, M. Andretta, M. Compiani, and G. Zanarini, Introduction to the Physics of Complex Systems, Pergamon Press, Oxford (1986).

    Google Scholar 

  12. R. M. Whitnell and K. R. Wilson, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd (Eds.), Vol. IV, VCH Publishers, Inc., New York, NY (1993).

    Google Scholar 

  13. E. Pollak, H. Grabert, and P. Hänggi, J. Chem. Phys. 91, 4073 (1989).

    Article  CAS  Google Scholar 

  14. I. Rips and E. Pollak, Phys. Rev. A 41, 5336 (1990).

    Article  Google Scholar 

  15. I. Rips, Phys. Rev. A 42, 4427 (1990).

    Article  CAS  Google Scholar 

  16. A. M. Frishman and E. Pollak, J. Chem. Phys. 98, 9532 (1993).

    Article  CAS  Google Scholar 

  17. G. R. Haynes, G. A. Voth, and E. Pollak, Chem. Phys. Lett. 207, 309 (1993).

    Article  Google Scholar 

  18. H. Eyring, J. Chem. Phys. 3, 107 (1935).

    Article  CAS  Google Scholar 

  19. E. P. Wigner, Trans. Faraday Soc. 34, 29 (1938).

    Article  CAS  Google Scholar 

  20. D. Chandler, J. Chem. Phys. 68, 2959 (1978).

    Article  CAS  Google Scholar 

  21. R. Kubo, in 1965 Tokyo Summer Lecture Series in Theoretical Physics. Part I. Many Body Theory, R. Kubo (Ed.), Syokabo/Benjamin, Tokyo/New York (1966).

    Google Scholar 

  22. R. Kubo, Rep. Prog. Phys. 29, 255 (1966).

    Article  CAS  Google Scholar 

  23. H. Mori, Prog. Theor. Phys. (Kyoto) 33, 423 (1965).

    Article  Google Scholar 

  24. B. J. Berne and R. Pecora, Dynamic Light Scattering, Wiley-Interscience, New York, NY (1976).

    Google Scholar 

  25. A. O. Caldeira and A. J. Leggett, Ann. of Phys. 149, 374 (1983).

    Article  Google Scholar 

  26. R. Zwanzig, J. Stat. Phys. 9, 215 (1973).

    Article  Google Scholar 

  27. G. W. Ford, M. Kac, and P. Mazur, J. Math. Phys. 6, 504 (1965).

    Article  Google Scholar 

  28. E. Cortes, B. J. West, and K. Lindenberg, J. Chem. Phys. 82, 2708 (1985).

    Article  CAS  Google Scholar 

  29. B. J. Gertner, J. P. Bergsma, K. R. Wilson, S. Lee, and J. T. Hynes, J. Chem. Phys. 86, 1377 (1987).

    Article  CAS  Google Scholar 

  30. E. Pollak, J. Chem. Phys. 86, 3944 (1987).

    Article  CAS  Google Scholar 

  31. S. C. Tucker, J. Phys. Chem. 97, 1596 (1993).

    Article  CAS  Google Scholar 

  32. S. E. Huston, P. J. Rossky, and D. A. Zichi, J. Amer. Chem. Soc. 111, 5680 (1989).

    Article  CAS  Google Scholar 

  33. P. Hänggi and F. Mojtabai, Phys. Rev. A 26, 1168 (1982).

    Article  Google Scholar 

  34. E. Pollak, J. Chem. Phys. 85, 865 (1986).

    Article  CAS  Google Scholar 

  35. Yu. I. Dakhnovskii and A. A. Ovchinnikov, Phys. Lett. 113A, 147 (1985).

    Google Scholar 

  36. S. H. Northrup and J. T. Hynes, J. Chem. Phys. 73, 2700 (1980).

    Article  CAS  Google Scholar 

  37. We identify Cv(t) with C22(t) of the original Grote-Hynes paper, [8].

    Google Scholar 

  38. J. E. Straub, M. Borkovec, and B. J. Berne, J. Phys. Chem. 91, 4995 (1987).

    Article  CAS  Google Scholar 

  39. R. F. Grote, G. van der Zwan, and J. T. Hynes, J. Phys. Chem. 88, 4676 (1984).

    Article  CAS  Google Scholar 

  40. J. P. Bergsma, J. R. Reimers, K. R. Wilson, and J. T. Hynes, J. Chem. Phys. 85, 5625 (1986).

    Article  CAS  Google Scholar 

  41. W. P. Keirstead, K. R. Wilson, and J. T. Hynes, J. Chem. Phys. 95, 5256 (1991).

    Article  CAS  Google Scholar 

  42. B. J. Gertner, K. R. Wilson, and J. T. Hynes, J. Chem. Phys. 90, 3537 (1989).

    Article  CAS  Google Scholar 

  43. J. P. Bergsma, B. J. Gertner, K. R. Wilson, and J. T. Hynes, J. Chem. Phys. 86, 1356 (1987).

    Article  CAS  Google Scholar 

  44. B. Roux and M. Karplus, J. Phys. Chem. 95, 4856 (1991).

    Article  CAS  Google Scholar 

  45. J. E. Straub, M. Borkovec, and B. J. Berne, J. Chem. Phys. 89, 4833 (1988).

    Article  CAS  Google Scholar 

  46. D. A. Zichi, G. Ciccotti, J. T. Hynes, and M. Ferrario, J. Phys. Chem. 93, 6261 (1989).

    Article  CAS  Google Scholar 

  47. G. Ciccotti, M. Ferrario, J. T. Hynes, and R. Kapral, J. Chem. Phys. 93, 7137 (1990).

    Article  CAS  Google Scholar 

  48. S. C. Tucker, M. E. Tuckerman, B. J. Berne, and E. Pollak, J. Chem. Phys. 95, 5809 (1991).

    Article  CAS  Google Scholar 

  49. J. B. Straus, J. M. G. Llorente, and G. A. Voth, J. Chem. Phys. 98, 4082 (1993).

    Article  CAS  Google Scholar 

  50. J. B. Straus and G. A. Voth, J. Chem. Phys. 96, 5460 (1992).

    Article  CAS  Google Scholar 

  51. E. Pollak, S.C. Tucker, and B. J. Berne, Phys. Rev. Lett. 65, 1399 (1990).

    Article  CAS  Google Scholar 

  52. E. Pollak, J. Chem. Phys. 93, 1116 (1990).

    Article  CAS  Google Scholar 

  53. J. C. Keck, Adv. Chem. Phys. 13, 85 (1967).

    Article  Google Scholar 

  54. J. B. Anderson, J. Chem. Phys. 58, 4684 (1973).

    Article  CAS  Google Scholar 

  55. B. H. Mahan, J. Chem. Educ. 51, 709 (1974).

    Article  CAS  Google Scholar 

  56. W. H. Miller, J. Chem. Phys. 61, 1823 (1974).

    Article  CAS  Google Scholar 

  57. B. C. Garrett and D. G. Truhlar, J. Phys. Chem. 83, 1052, 1079 (1979); ibid. 87, 4553(E) (1983).

    Article  CAS  Google Scholar 

  58. S.C. Tucker and D. G. Truhlar, in New Theoretical Concepts for Understanding Organic Reactions, NATO ASI Ser. C, J. Bertrán and I. G. Csizmadia (Eds.), Kluwer Academic, Dordrecht (1989), provides a recent review of multidimensional transition state theory.

    Google Scholar 

  59. D. G. Truhlar, A. D. Isaacson, and B. C. Garrett, in Theory of Chemical Reaction Dynamics, M. Baer (Ed.), Vol. 4, CRC Press, Boca Raton, FL (1985).

    Google Scholar 

  60. D. G. Truhlar and B. C. Garrett, Ann. Rev. Phys. Chem. 35, 159 (1984).

    Article  CAS  Google Scholar 

  61. P. Pechukas, in Dynamics of Molecular Collisions, Part B., W. H. Miller (Ed.), Plenum Press, New York, NY (1976).

    Google Scholar 

  62. E. Pollak, in Theory of Chemical Reaction Dynamics, M. Baer (Ed.), Vol. 3, CRC Press, Boca Raton, FL (1985).

    Google Scholar 

  63. P. Pechukas, Ann. Rev. Phys. Chem. 32, 159 (1981).

    Article  CAS  Google Scholar 

  64. B. C. Garrett and D. G. Truhlar, J. Chem. Phys. 70, 1593 (1979).

    Article  CAS  Google Scholar 

  65. E. Pollak and P. Pechukas, J. Chem. Phys. 69, 1218 (1978).

    Article  CAS  Google Scholar 

  66. P. Pechukas and E. Pollak, J. Chem. Phys. 71, 2062 (1979).

    Article  CAS  Google Scholar 

  67. B. J. Berne, in Multiple Time Scales, J. U. Brackbill and B. I. Cohen (Eds.), Academic Press, New York, NY (1985).

    Google Scholar 

  68. E. Pollak and P. Talkner, Phys. Rev. E 47, 922 (1993).

    Article  Google Scholar 

  69. E. Pollak, Mod. Phys. Lett. B 5, 13 (1991).

    Article  CAS  Google Scholar 

  70. A. Frishman and E. Pollak, J. Chem. Phys. 96, 8877 (1992).

    Article  CAS  Google Scholar 

  71. D. A. McQuarrie, Statistical Mechanics, Harper and Row, New York, NY (1976).

    Google Scholar 

  72. G. van der Zwan and J. T. Hynes, J. Chem. Phys. 78, 4174 (1983).

    Article  Google Scholar 

  73. A. M. Berezhkovskii, E. Pollak, and V. Y. Zitserman, J. Chem. Phys. 97, 2422 (1992).

    Article  CAS  Google Scholar 

  74. A. M. Levine, M. Shapiro, and E. Pollak, J. Chem. Phys. 88, 1959 (1988).

    Article  CAS  Google Scholar 

  75. E. Pollak, J. Chem. Phys. 95, 533 (1991).

    Article  CAS  Google Scholar 

  76. This expression for the dividing surface does preclude dividing surfaces which are independent of p.11. E. Pollak, personal communication.

    Google Scholar 

  77. H. J. Kim and J. T. Hynes, J. Chem. Phys. 96, 5088 (1992).

    Article  CAS  Google Scholar 

  78. S. Lee and J. T. Hynes, J. Chem. Phys. 88, 6853, 6863 (1988).

    Article  Google Scholar 

  79. H. J. Kim and J. T Hynes, J. Amer. Chem. Soc. 114, 10528 (1992).

    Article  CAS  Google Scholar 

  80. D. G. Truhlar, G. K. Schenter, and B. C. Garrett, J. Chem. Phys. 98, 5756 (1993).

    Article  CAS  Google Scholar 

  81. H. J. Kim and J. T. Hynes, J. Amer. Chem. Soc. 114, 10508 (1992).

    Article  CAS  Google Scholar 

  82. E. Pollak, J. Phys. Chem. 95, 10235 (1991).

    Article  CAS  Google Scholar 

  83. While the reactant partition function integral is technically bounded by the dividing surface, it is generally assumed that the integral is converged before this boundary is reached, and thus that the reactant partition function is independent of the dividing surface. This will be a good assumption for high barrier reactions.

    Google Scholar 

  84. QR is here the reactant partition function for the two-dimensional free energy Hamiltonian, as in Equation (50).

    Google Scholar 

  85. The prefactor differs from that in the original reference becuase here we consider the unnormalized rate, QRk2dTST, instead of the ratio P = k2dTST/kGH.

    Google Scholar 

  86. J. E. Straub, B. J. Berne, and B. Roux, J. Chem. Phys. 93, 6804 (1990).

    Article  CAS  Google Scholar 

  87. G. A. Voth, J. Chem. Phys. 97, 5908 (1992).

    Article  CAS  Google Scholar 

  88. S. Singh, R. Krishnan, and G. W. Robinson, Chem. Phys. Lett. 175, 338 (1990).

    Article  CAS  Google Scholar 

  89. R. Krishnan, S. Singh, and G. W. Robinson, Phys. Rev. A 45, 4508 (1992).

    Article  Google Scholar 

  90. R. Krishnan, S. Singh, and G. W. Robinson, J. Chem. Phys. 97, 5516 (1992).

    Article  CAS  Google Scholar 

  91. By exact we mean that the reduced-dimensionality expression is exactly equivalent to the multidimensional expression with the same dividing surface.

    Google Scholar 

  92. S. C. Tucker and E. Pollak, J. Stat. Phys. 66, 975 (1992).

    Article  Google Scholar 

  93. In the original paper, [93], the n + 1-dimensonal flux expression, which must be normalized by the reactants configurational partition function in the full dimensionality space, was given. Here, Equation (79) is already properly normalized by QR [85].

    Google Scholar 

  94. See for example [24].

    Google Scholar 

  95. G. K. Schenter, R. P. McRae, and B. C. Garrett, J. Chem. Phys. 97, 9116 (1992).

    Article  CAS  Google Scholar 

  96. See references in [73].

    Google Scholar 

  97. A. M. Berezhkovskii and V. Y. Zitserman, Chem. Phys. Lett. 172, 235 (1990).

    Article  CAS  Google Scholar 

  98. A. M. Berezhkovskii and V. Y. Zitserman, Chem. Phys. 157, 141 (1991).

    Article  CAS  Google Scholar 

  99. J. S. Langer, Ann. Phys. (N.Y.) 54, 258 (1969).

    Article  CAS  Google Scholar 

  100. M. Smoluchowski, Z. Phys. Chem. 92, 129 (1918).

    Google Scholar 

  101. R. F. Grote and J. T. Hynes, J. Chem. Phys. 74, 4465 (1981).

    Article  CAS  Google Scholar 

  102. R. F. Grote and J. T. Hynes, J. Chem. Phys. 75, 2191 (1981).

    Article  CAS  Google Scholar 

  103. A. Nitzan, Adv. Chem. Phys. 70, 489 (1988).

    Article  Google Scholar 

  104. A. Nitzan, J. Chem. Phys. 86, 2734 (1987).

    Article  CAS  Google Scholar 

  105. Application of the BPZ formalism requires choosing a solute coordinate representation in which the friction tensor is diagonal.

    Google Scholar 

  106. S. K. Reese, S. C. Tucker, and G. K. Schenter, work in progress.

    Google Scholar 

  107. C. A. Parr and D. G. Truhlar, J. Phys. Chem. 75, 1844 (1971).

    Article  Google Scholar 

  108. S. C. Tucker and D. G. Truhlar, J. Amer. Chem. Soc. 112, 3347 (1990).

    Article  CAS  Google Scholar 

  109. R. P. McRae, G. K. Schenter, B. C. Garrett, G. R. Haynes, G. A. Voth, and G. C. Schatz, J. Chem. Phys. 97, 7392 (1992).

    Article  CAS  Google Scholar 

  110. M. J. Gillan, J. Phys. C 20, 3621 (1987).

    Article  Google Scholar 

  111. G. A. Voth, D. Chandler, and W. H. Miller, J. Chem. Phys. 91, 7749 (1989).

    Article  CAS  Google Scholar 

  112. G. A. Voth, Chem. Phys. Lett. 170, 289 (1990).

    Article  CAS  Google Scholar 

  113. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, NY (1965).

    Google Scholar 

  114. L. S. Schulman, Techniques and Applications of Path Integrals, John Wiley & Sons, New York, NY (1981).

    Google Scholar 

  115. J. J. Sakurai, Modern Quantum Mechanics, Benjamin/Cummings, Menlo Park, CA (1985).

    Google Scholar 

  116. M. Messina, G. K. Schenter, and B. C. Garrett, J. Chem. Phys. 98, 8525 (1993).

    Article  CAS  Google Scholar 

  117. The idea of using a quantum distribution function is not new, see [56].

    Google Scholar 

  118. J. D. Doll and D. L. Freeman, Adv. Chem. Phys. 73, 289 (1989).

    Article  Google Scholar 

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Authors
  1. Susan C. Tucker
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Editors and Affiliations

  1. Paul Scherrer Institute, Villingen, Switzerland

    Peter Talkner

  2. Department of Physics, Universität Augsburg, Augsburg, Germany

    Peter Hänggi

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Tucker, S.C. (1995). Variational Transition State Theory in Condensed Phases. In: Talkner, P., Hänggi, P. (eds) New Trends in Kramers’ Reaction Rate Theory. Understanding Chemical Reactivity, vol 11. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0465-4_2

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  • DOI: https://doi.org/10.1007/978-94-011-0465-4_2

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