Centroid density transition-state theory

Warshel and Chu [42] and Hwang et al. [60] were the first to calculate the contribution of tunneling and other nuclear quantum effects to PT in solution and enzyme catalysis, respectively. Since then, and in particular in the past few years, there has been a significant increase in simulations of quantum mechanical-nuclear effects in enzyme and in solution reactions [16]. The approaches used range from the quantized classical path (QCP) (for example. Refs. [4, 58, 95]), the centroid path integral approach [54, 55], and variational transition state theory [96], to the molecular dynamics with quantum transition (MDQT) surface hopping method [31] and density matrix evolution [97-99]. Most studies of enzymatic reactions did not yet examine the reference water reaction, and thus could only evaluate the quantum mechanical contribution to the enzyme rate constant, rather than the corresponding catalytic effect. However, studies that explored the actual catalytic contributions (for example. Refs. [4, 58, 95]) concluded that the quantum mechanical contributions are similar for the reaction in the enzyme and in solution, and thus, do not contribute to catalysis. 

One conclusion that can be reached from the early work on effective potentials [1,21-23], the work of Cao and Voth [3-8], as well as the centroid density-based formulation of quantum transition-state theory [42-44,49] is that the path centroid is a particularly useful variable in statistical mechanics about which to develop approximate, but quite accurate, quantum mechanical expressions and to probe the quantum-classical correspondence principle. It is in this spirit that a general centroid density-based formulation of quantum Boltzmann statistical mechanics is presented in the present section. This topic is the subject of Paper I, and the emphasis in this section is on analytic theory as opposed to computational approaches (cf. Sections III and IV). 

As a direct test of the analytic theory for the centroid density, the quantum correction factor for the thermal rate constant of an Eckart barrier potential was calculated within the context of path-integral quantum transition-state theory [42-44,49]. The results are tabulated in... 

Figure 7 Quantum transmission coefficient for a symmetric double well potential interacting with a generic dissipative medium as a function of friction strength y. The barrier height is denoted b and (Ob is the imaginary frequency at the top of the potential barrier. Solid squares QUAPI results. Dashed line results of centroid-density quantum transition state theory, (a) Eb/kbT = 10 (activated regime with tunneling contributions), (b) Eb/kbT = 30 for the first six data points (at small friction) the temperature is below crossover. Data from Ref. 51...

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