Transition State Theory

Transition State Theory is a classical theory developed by Wigner, Eyring, Gwynne Evans and Polanyi, [1], [2], [3], [4] that calculates the rate of the reaction as the equilibrium flux of reactive trajectories through the DS. Although there has been much discussion of quantum mechanical analogs of TST [5], [6] , we will briefly discuss in this book the original version based on classical mechanics, since the dividing surface concept, that we use, is incompatible with quantum mechanics. The overview that we give is not supposed to be a review of the vast literature that exists in this area, but to connect the mathematical study of the geometrical structures that characterize reaction dynamics in phase space.

It is commonly claimed that conventional TST makes two main assumptions. [7]

• The first, called the equilibrium assumption, requires that the reactant state and TS be in thermal equilibrium. The maintenance of energetic equilibrium means that the thermalization maintaining this equilibrium is (at least) as fast as the rate at which these states are depopulated. [8], [9] The equilibrium condition is usually satisfied for most gas-phase bimolecular reactions and for reactions in the liquid phase, because energy exchange between solutes and solvent is usually rapid enough to maintain the equilibrium.[10], [11] However, there are cases where equilibrium is not maintained, even in solution.[12] In addition, for unimolecular reactions of intermediates with low barriers to product formation, it is commonly the case that most trajectories coming from the reactant state will have enough energy in a product-forming reaction coordinate to cross the second barrier as soon as they reach it.[13], [14]

• The second claimed assumption specifies that any trajectory crossing the TS dividing surface from the reactant state is on a path towards the product state and will reach it without recrossing the dividing surface prior to the product being reached.[15] Because the rate is calculated as the flux through the TS, any non-reactive trajectory that crosses the TS dividing surface, or reactive trajectory that crosses it more than once will increase the flux through the dividing surface, thus leading to an overestimate of the rate constant. This means that TST gives us an upper limit on the true rate constant, and that if we found a dividing surface without any recrossing then TST would give the exact value of the rate constant (subject to certain caveats[16]).

In conventional TST the transition state dividing surface is located at the saddle point, which is the maximum energy point on the minimum energy path from reactants to products. However, TST is most powerful in the form of Variational Transition State Theory (VTST), [8], [9], [15], [17], [18], [19], [20], [21] which is a generalization of TST that removes the restriction on the dividing surface to cross the saddle point. In VTST the dividing surface is variationally optimized to minimize the rate constant, usually by finding the maximum free energy along the reaction path. Although this surface is properly located in phase space, most of the VTST calculations assume that the TS can be found in configuration space.[13]

References

1. H. Pelzer and E. Wigner, “The speed constansts of the exchange reactions,” Z. Phys. Chem. B., vol. 15, p. 445, 1932.
2. H. Eyring, “The Activated Complex in Chemical Reactions,” J. Chem. Phys., vol. 3, no. 2, pp. 107–115, 1935.
3. M. G. Evans and M. Polanyi, “Some applications of the transition state method to the calculation of reaction velocities, especially in solution,” Trans. Far. Soc., vol. 31, no. 0, pp. 875–894, 1935.
4. S. Glasstone, K. J. Laidler, and H. Eyring, The theory of rate processes: the kinetics of chemical reactions, viscosity, diffusion and electrochemical phenomena. New York: McGraw-Hill Book Company, inc., 1941.
5. P. Pechukas, “Recent Developments in Transition State Theory,” Ber. Bunsen-Ges. Phys. Chem., vol. 86, no. 5, pp. 372–378, 1982.
6. H. Waalkens, R. Schubert, and S. Wiggins, “Wigner’s dynamical transition state theory in phase space: classical and quantum,” Nonlinearity, vol. 21, no. 1, pp. R1–R118, 2007.
7. B. H. Mahan, “Activated complex theory of bimolecular reactions,” J. Chem. Ed., vol. 51, no. 11, p. 709, 1974.
8. J. L. Bao and D. G. Truhlar, “Variational transition state theory: theoretical framework and recent developments,” Chem. Soc. Rev., vol. 46, no. 24, pp. 7548–7596, 2017.
9. B. C. Garrett and D. G. Truhlar, “Generalized transition state theory. Classical mechanical theory and applications to collinear reactions of hydrogen molecules,” J. Phys. Chem., vol. 83, no. 8, pp. 1052–1079, 1979.
10. J. B. Anderson, “Statistical theories of chemical reactions. Distributions in the transition region,” J. Chem. Phys., vol. 58, no. 10, pp. 4684–4692, 1973.
11. J. B. Anderson, “Predicting Rare Events in Molecular Dynamics,” in Advances in Chemical Physics, 1995.
12. S. Essafi and J. N. Harvey, “Rates of Molecular Vibrational Energy Transfer in Organic Solutions,” J. Phys. Chem. A, vol. 122, no. 14, pp. 3535–3540, 2018.
13. B. K. Carpenter, “Trajectories through an intermediate at a fourfold branch point. Implications for the stereochemistry of biradical reactions,” J. Am. Chem. Soc., vol. 107, no. 20, pp. 5730–5732, 1985.
14. P. Collins, Z. C. Kramer, B. K. Carpenter, G. S. Ezra, and S. Wiggins, “Nonstatistical dynamics on the caldera,” J. Chem. Phys., vol. 141, no. 3, p. 034111, 2014.
15. D. G. Truhlar and B. C. Garrett, “Variational transition-state theory,” Acc. Chem. Res., vol. 13, no. 12, pp. 440–448, 1980.
16. F. A. L. Mauguiere et al., “Phase space barriers and dividing surfaces in the absence of critical points of the potential energy: Application to roaming in ozone,” J. Chem. Phys., vol. 144, no. 5, p. 054107, 2016.
17. D. G. Truhlar and B. C. Garrett, “Variational Transition State Theory,” Annu. Rev. Phys. Chem., vol. 35, no. 1, pp. 159–189, 1984.
18. B. C. Garrett and D. G. Truhlar, “Criterion of minimum state density in the transition state theory of bimolecular reactions,” J. Chem. Phys., vol. 70, no. 4, pp. 1593–1598, 1979.
19. J. C. Keck, “Variational Theory of Chemical Reaction Rates Applied to Three‐Body Recombinations,” J. Chem. Phys., vol. 32, no. 4, pp. 1035–1050, 1960.
20. J. Horiuti, “On the Statistical Mechanical Treatment of the Absolute Rate of Chemical Reaction,” Bull. Chem. Soc. Jpn., vol. 13, no. 1, pp. 210–216, 1938.
21. E. Wigner, “Calculation of the Rate of Elementary Association Reactions,” J. Chem. Phys., vol. 5, no. 9, pp. 720–725, 1937.