Quantum transition-state theory

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. 

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. 

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Voth G A, Chandler D and Miller W H 1989 Rigorous formulation of quantum transition state theory and its dynamical corrections J. Chem. Phys. 91 7749... 

Makarov D E and Topaler M 1995 Quantum transition-state theory below the crossover temperature Phys. Rev. E 52 178... 

Stuchebrukhov A A 1991 Green s functions in quantum transition state theory J. Chem. Phys. 95 4258... 

Shao J, Liao J-L and Poliak E 1998 Quantum transition state theory—perturbation expansion J. Chem. Phys. 108 9711 Liao J-L and Poliak E 1999 A test of quantum transition state theory for a system with two degrees of freedom J. 

According to the quantum transition state theory [108], and ignoring damping, at a temperature T h(S) /Inks — a/ i )To/2n, the wall motion will typically be classically activated. This temperature lies within the plateau in thermal conductivity [19]. This estimate will be lowered if damping, which becomes considerable also at these temperatures, is included in the treatment. Indeed, as shown later in this section, interaction with phonons results in the usual phenomena of frequency shift and level broadening in an internal resonance. Also, activated motion necessarily implies that the system is multilevel. While a complete characterization of all the states does not seem realistic at present, we can extract at least the spectrum of their important subset, namely, those that correspond to the vibrational excitations of the mosaic, whose spectraFspatial density will turn out to be sufficiently high to account for the existence of the boson peak. 

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... 

Zhao, M. and Rice, S. A. Resonance state approach to quantum transition state theory, J. Phys. Chem., 98 (1994), 3444-2449... 

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.)...

V.2.2 Quantum transition state theory. The centroid method is one way... 

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 " ... 

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. 

G. Mills, G. K. Schenter, D. Makarov and H. Jonsson, RAW Quantum Transition State Theory , in Classical and Quantum Dynamics in Condensed Phase Simulations , ed. B. J. Berne, G. Ciccotti and D. F. Coker, page 405 (World Seientific, 1998). 

Several approaches to the issue of tight binding ionic mobility in the channel system can be explored. In the past decade, quantum transition state theory has matured to the point that it is possible to consider sophisticated treatments that include formally accurate... 

VoTH, G. A., Chandler, D., Miller, W. H., Rigorous Formulation of Quantum Transition State Theory and Its Dynamical Corrections, J. Chem Phys. 1989, 91, 7749. 

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