Quantized classical path method

Several other path integral-based approaches to compute KIE exist, which however do not use QI approximation. These include, for example, approaches using other quantum TSTs [55-60] or the quantized classical path method [7,61,62]. 

Hwang et al.131 were the first to calculate the contribution of tunneling and other nuclear quantum effects to enzyme catalysis. Since then, and in particular in the past few years, there has been a significant increase in simulations of QM-nuclear effects in enzyme reactions. The approaches used range from the quantized classical path (QCP) (e.g., Refs. 4,57,136), the centroid path integral approach,137,138 and vibrational TS theory,139 to the molecular dynamics with quantum transition (MDQT) surface hopping method.140 Most studies did not yet examine the reference water reaction, and thus could only evaluate the QM contribution to the enzyme rate constant, rather than the corresponding catalytic effect. However, studies that explored the actual catalytic contributions (e.g., Refs. 4,57,136) concluded that the QM contributions are similar for the reaction in the enzyme and in solution, and thus, do not contribute to catalysis. 

On the other hand, the obvious concern about this method is inaccuracy arising from the frozen Gaussian exponents (see below in Sec. 4.3.5 though). It is desirable to seek for a theory more satisfactory to quantize non-classical path in the quality level similar to or better than that of the WKB theory. 

The classical GLE method can be combined with a quantum mechanical description of the gas atom or molecule [87]. The two systems are coupled together by letting the surface atoms be forced by a mean field potential obtained by averaging over the gas atom coordinates [89]. Such models are often referred to classical path models. Other classical path models treat the gas atom dynamics by classical mechanics, whereas the phonons are quantized. 

About 50 years after Einstein, Gutzwiller applied the path integral method with a semiclassical approximation and succeeded to derive an approximate quantization condition for the system that has fully chaotic classical counterpart. His formula expresses the density of states in terms of unstable periodic orbits. It is now called the Gutzwiller trace formula [9,10]. In the last two decades, several physicists tested the Gutzwiiler trace formula for various... 

Ion-molecule association is seemingly well suited for the apphcation of the QCT method. Since there is no potential barrier and the centrifugal potential is broad, quantum mechanical turmeling, which is an issue with the QCT method, is typically im-important. Energy transfer from relative translational to vibrational and/or rotational motions of the ion-molecule complex should be reasonably classical because of the large density of states involved. Additionally, since the variational transition state has an early location along the reaction path, quantization of reactant vibrational motions should result in a reasonably correct treatment of these motions at the TS. 

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