Abstract
As one tries to construct an increasingly rigorous quantum mechanical generalization of classical transition state theory, one that is free of all ‘extraneous’ approximations (e.g., separability of a one dimensional reaction coordinate), one is ultimately driven to the dynamically exact quantum treatment. Though it seems pointless to call this a transition state ‘theory’ (it is in effect a quantum mechanical simulation), it is nevertheless possible using transition state-like ideas to cast a fully rigorous quantum approach in a form that allows one to carry out such calculations without having to solve the complete state-to-state quantum reactive scattering problem. Rigorous calculations for the reactions H + H2 → H2 + H, H + O2 → OH + O, and H + H2O → H2 + OH illustrate this approach. At the semiclassical level, there does exist a version of transition state theory — based on the locally ‘good’ action variables about the saddle point on the potential energy surface — which includes non-separable coupling between all degrees of freedom (including the reaction coordinate) in a unified manner.
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References
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N(E) will always increase monotonically with E in a transition state approximation. It is easy to prove this classically, e.g., from Equation (15). If the dividing surface is held fixed as E varies, then from Equation (15) one has \(\frac{d}{{dE}}{{N}_{{TST}}}(E) = {{(2\pi \hbar )}^{{ - (F - 1)}}}\int {dp\prime } \int {dq\prime \delta [{\rm E} - {\rm H}\ddag ],}\) which is clearly positive. Furthermore, if the dividing surface is parameterized and allowed to vary with energy, the above equation still holds because any parameters in the dividing surface are chosen variationally. Thus if the dividing surface (and thus the Hamil-tonian H‡) depend on some parameters c 1, c 2,... = {c k }, then the expression for N TST will depend not only on the energy E but also on these parameters, N TST(E, c 1, c 2,...). The values of the c k ’s are chosen, however, by the variational condition \( 0 = {\partial \over {\partial {c_k}}}{N_{TST}}(E,\{ {c_k}\} ), \)which determines specific values c k (E). Thus the variationally optimized result for the cumulative reaction probability is N TST(E, c 1(E), c 2(E),...) ≡ N TST(E). Then \( {d \over {dE}}{N_{TST}}(E) = {\partial \over {\partial E}}{N_{TST}}(E\{ {c_k}\} ) + \sum\limits_k {{{\partial {N_{TST}}} \over {\partial {c_k}}}} {\rm{ }}{c'_k}(E), \) but the last terms are all zero because of the variational conditions.
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Miller, W.H. (1995). Recent Developments in the Quantum Mechanical Theory of Chemical Reaction Rates. 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_11
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DOI: https://doi.org/10.1007/978-94-011-0465-4_11
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