Quantum transitional state theory QTST

We begin our discussion with path integral quantum transition state theory (QTST) [14], which is the theoretical model that we use to model enzymatic reactions. In QTST, the exact rate constant is expressed by the QTST rate constant, qtst, multiplied by a transmission coefficient yq ... 

As described below, it is possible to construct a theory which satisfies conditions a-d and at least thus far it has been found empirically to bound the exact quantum rate from above. This Quantum Transition State Theory (QTST) is predicated on the exact quantum expression for the reactive flux, derived by Miller, Schwartz and Trompi " ... 

Since transition state theory is derived from a classical flux correlation function, it has all shortcomings of a classical description of the reaction process. Neither tunneling, which is especially important for H-atom transfer processes or low temperature reactions, nor zero point energy effects are included in the description. Thus, the idea to develop a quantum transition state theory (QTST) which accounts for quantum effects but retains the computational advantages of the transition state approximation has been very attractive (for examples see Refs.[5, 6] and references therein). The computation of these QTST rate constants does not require the calculation of real-time dynamics and is therefore feasible for large molecular systems. 

However, there are fundamental problems in the derivation of a quantum transition state theory. TST requires the simultaneous knowledge of position and momentum the direction of the initial momentum at the dividing surface is a key ingredient to the theory. Thus, TST violates the uncertainty principle and a straightforward derivation of a quantum transition state theory is not possible. Ad hoc assumptions are required in the introduction of a QTST. Truhlar and coworkers, for example, introduce a specific one-dimensional path and add a tunneling correction, calculated along this path, to account for quantum effects in transition state theory calculations. Poliak and coworkers employ a harmonic approximation at the saddle point to obtain a quantum approximation for the dynamial factor. 

For another perspective we mention a second approach of which the reader should be aware. In this approach the dividing surface of transition state theory is defined not in terms of a classical mechanical reaction coordinate but rather in terms of the centroid coordinate of a path integral (path integral quantum TST, or PI-QTST) [96-99] or the average coordinate of a quanta wave packet. In model studies of a symmetric reaction, it was shown that the PI-QTST approach agrees well with the multidimensional transmission coefScient approach used here when the frequency of the bath is high, but both approaches are less accurate when the frequency is low, probably due to anharmonicity [98] and the path centroid constraint [97[. However, further analysis is needed to develop practical PI-QTST-type methods for asymmetric reactions [99]. 

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