Path-integral quantum transition-state

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-... 

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]. 

The reactive flux method is also useful in calculating rate constants in quantum systems. The path integral formulation of the reactive flux together with the use of the centroid distribution function has proved very useful for the calculation of quantum transition-state rate constants [7]. In addition new methods, such as the Meyer-Miller method [8] for semiclassical dynamics, have been used to calculate the flux-flux correlation function and the reactive flux. 

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... 

In this section we review several studies of phase transitions in adsorbed layers. Phase transitions in adsorbed (2D) fluids and in adsorbed layers of molecules are studied with a combination of path integral Monte Carlo, Gibbs ensemble Monte Carlo (GEMC), and finite size scaling techniques. Phase diagrams of fluids with internal quantum states are analyzed. Adsorbed layers of H2 molecules at a full monolayer coverage in the /3 X /3 structure have a higher transition temperature to the disordered phase compared to the system with the heavier D2 molecules this effect is... 

Feynman built on this work and in 1948 it culminated in his path integral formulation of quantum mechanics. In the classical limit considered by EXirac, only one trajectory connects the system at time tj to that at time t2 and he limited his discussion to this case. What Feynman did was to consider all the trajectories or paths that connect the states at the initial and final times, since he wished to obtain the corresponding quantum limit. Each path has its own value for the action W and all the values of exp[(i/ft)lF] must be added together to obtain the total transition amplitude. Thus, the expression for the transition amplitude between the states and 1, 2> is th su of th elementary contributions, one from each trajectory passing between q,i at time ti and q,2 at time tj. Each of these contributions has the same modulus, but its phase is the classical action integral (l/fi) Ldt for the path. This is expressed as. 

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