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Classical transition state theory is exact if the transition state is unique

J. Chem. Phys. 71, 2062 (1979); doi:10.1063/1.438575

Issue Date: 1 September 1979

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Philip Pechukas and Eli Pollak
Department of Chemistry, Columbia University, New York, New York 10027
Department of Chemical Physics, Weizmann Institute of Science, Rehovoth, Israel
Under mild conditions on the long-range behavior of the potential, we show that classical transition state theory is exact at energy E for a collinear atom–diatom reaction if there is only one candidate for transition state at energy E—that is, only one periodic vibration of energy E across the interaction region. The Journal of Chemical Physics is copyrighted by The American Institute of Physics.
Permalink: http://link.aip.org/link/?JCPSA6/71/2062/1
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KEYWORDS and PACS

Keywords
PACS
  • 82.20.Db
    Physical chemistry Chemical kinetics Statistical theories, including transition state
  • YEAR: 1979

PUBLICATION DATA

ISSN:
0021-9606 (print)   1089-7690 (online)
Publisher:
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REFERENCES (9)
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  1. E. Wigner, J. Chem. Phys. 5, 720 (1937);
    2. Trans. Faraday Soc. 34, 29 (1938).
  2. See especially J. Horiuti, Bull. Chem. Soc. Jpn. 13, 210 (1938);
    3. J. C. Keck, Adv. Chem. Phys. 13, 85 (1967);
    G. W. Koeppl, J. Am. Chem. Soc. 96, 6539 (1974);
    P. Pechukas, in Dynamics of Molecular Collisions, Part B, edited by W. H. Miller (Vol. 2 of Modern Theoretical Chemistry) (Plenum, New York, 1976), Chap. 6;
    B. C. Garrett and D. G. Truhlar, J. Phys. Chem. 83, 1052 (1979).
  3. E. Pollak and P. Pechukas, J. Chem. Phys. 69, 1218 (1978).
  4. See also D. I. Sverdlik and G. W. Koeppl, Chem. Phys. Lett. 59, 449 (1978).
  5. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).
  6. E. Pollak and P. Pechukas, J. Chem. Phys. 70, 325 (1979).
  7. See, for instance, W. Rudin, Principles of Mathematical Analysis (McGraw-Hill, New York, 1953).
  8. The inverse (t,s) (q) is obviously well defined and continuous. but not differentiable, on the image of the larger region 0<=t<=t[prime], s1<=s<=s2.
  9. It is easy to see that q(tau) must in fact leave the region 0<t<t[prime], s1<s<s2 in finite time. Otherwise, the limit set of the trajectory q(tau) consists of points lying on the trajectory segment q(t,s[infinity]), 0<=t<=t[prime], where s[infinity] = lims(tau) as tau-->[infinity]; but that is impossible, because the limit set of a classical trajectory consists of complete classical trajectories, and q(t,s[infinity]), 0<=t<=t[prime], is not a complete classical trajectory since the velocity [partial-derivative]q/[partial-derivative]t does not vanish at t = t[prime].

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