Path-integral quantum transition-state theory

THEORETICAL BACKGROUND - PATH INTEGRAL QUANTUM TRANSITION STATE THEORY... 

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 ... 

Jang S, Voth GA (2001) A relationship between centroid dynamics and path integral quantum transition state theory. J Chem Phys 112(8747-8757) Erratum 114, 1944... 

Figure 11. The solid line depicts the quantum adiabatic free energy curve for the Fe /Fe electron transfer at the water/Pt(lll) interface (obtained by using the Anderson-Newns model, path integral quantum transition state theory, and the umbrella sampling of molecular dynamics. The dashed line shows the curve from the classical calculation as given in Fig. 5. (Reprinted from Ref 14.)...

Following Fey nman s original work, several authors pmsued extensions of the effective potential idea to construct variational approximations for the quantum partition function (see, e g., Refs. 7,8). The importance of the path centroid variable in quantum activated rate processes was also explored and revealed, which gave rise to path integral quantum transition state theory and even more general approaches. The Centroid Molecular Dynamics (CMD) method for quantum dynamics simulation was also formulated. In the CMD method, the position centroid evolves classically on the efiective centroid potential. Various analysis and numerical tests for realistic systems have shown that CMD captures the main quantum effects for several processes in condensed matter such as transport phenomena. 

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... 

G. A. Voth, J. Phys. Chem. 97, 8365 (1993). For a review of path integral quantum transition state theory, see this paper. 

M. J. Gillan, J. Phys. C 20, 3621 (1987). This paper is related to the path integral quantum transition state theory work of Refs. 42-44. 

The theoretical framework in the present discussion is path integral quantum transition state theory which is derived by writing the rate ex-... 

Quantum Transition State Theory and Path Integral Simulations... 

Voth G A 1993 Feynman path integral formulation of quantum mechanical transition state theory J. Phys. Chem. 97 8365... 

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]. 

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. 

Rick, S.W., Lynch, D.L., Doll, J.D. The quantum dynamics of hydrogen and deuterium on the Pd(III) surface A path integral transition state theory study, J. Chem. Phys. 1993, 99, 8183. 

In the second section the calculation of the rate constant was discussed from the classical mechanics viewpoint. Voth, Chandler, and Miller derived a quantum mechanical expression for the rate constant based on a path integral formalism. Using this expression as a starting point, Voth and O Gormani derived an effective barrier model to allow the calculation of the barrier tunneling contribution to the quantum mechanical rate constant for reactions in dissipative baths. The spirit of their derivation is quite similar to that which treats Grote-Hynes theory o as transition state theory for a parabolic barrier in a harmonic bath. 

If information on the reaction path is available, as, for instance, in variational transition state theory, this can be used to calculate k [69,70]. In transition state theory, only the knowledge of the energy and its first and second derivatives at the reactant and transition state locations is needed and the barrier is typically approximated by a simple functional form. One possibility is to describe the reaction barrier by an Eckart potential [75] (also called a sech potential, depending on the literature source), k in Eq. (7.19) is defined as the ratio of transmitted quantum particles to classical particles and the resulting integral for the Eckart potential can be solved numerically. An approximate solution is the Wigner tunneling correction ... 

W. H. Miller, Path integral representation of the reaction rate constant in quantum mechanical transition state theory, J. Chem. Phys. 63 1166 (1975). 

The path integral formulation of quantum theory provides a framework to describe the behavior of solvated electrons. Feynman used the approach to treat the slow moving electron in ionic crystals — the prototypical polaron problem. We have extended this theory, drawing on theories of the liquid state, to analyze the localization transition and related phenomena found with excess electrons in fluids. 

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