Quantum transition-state theory reactions

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 ... 

A. Dynamical Test of the Centroid Quantum Transition-State Theory for Electron Transfer Reactions... 

Applications are then presented in Section IV. These examples should served as a guide as to what kinds of problems can be studied with these techniques and the limitations and possibilities for these methods. We present three examples (1) a dynamical test of the centroid quantum transition-state theory for electron transfer (ET) reactions in the crossover regime between adiabatic and nonadiabatic electron transfer, (2) the primary electron transfer reaction in bacterial photosynthesis, and (3) the diffusion kinetics of a Brownian particle in a periodic potential. Finally, Section V offers an outlook and a perspective of the current status of the field from our vantage point. 

Figure 21. Adibatic solvent activation free-energy curves for the Fe /Fe electron transfer reaction with a platinum electrode at 300 K calculated obtained using the model of Ref. 50, path-integral quantum transition-state theory, and umbrella sampling. The solid line depicts the quantum adiabatic free-energy curve, while the dashed line depicts the curve in the classical limit. In both cases, the left-hand well corresponds to the Fe stable state, while the right-hand well is the Fe " stable state.

Due to the central role the reaction rate constant plays in physical chemistiy, many more or less accurate approximations for this quantity have been developed over time, starting from the Arrhenius equation [1] and transition state theory (TST) [2-4]. Among the most accurate of such approximations are so-called quantum transition state theories [5-18], which treat the rate constant quantum mechanically, but, similarly to the original classical TST, still rely on some sort of a transition state assumption. A recent such approximation that can also treat general many-dimensional systems is the quantum instanton (Ql) approximation of Miller et al. [17]. 

Transition State Theory Reaction dynamics can be treated quantum... 

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. 

Miller W H 1974 Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants J. Chem. Phys. 61 1823-34... 

Transition state theory assumes an equilibrium energy distribution among all possible quantum states at all points along the reaction coordinate. The probability of finding a molecule in a given quantum state is proportional to which is a Boltzmann... 

Quantum mechanical effects—tunneling and interference, resonances, and electronic nonadiabaticity— play important roles in many chemical reactions. Rigorous quantum dynamics studies, that is, numerically accurate solutions of either the time-independent or time-dependent Schrodinger equations, provide the most correct and detailed description of a chemical reaction. While hmited to relatively small numbers of atoms by the standards of ordinary chemistry, numerically accurate quantum dynamics provides not only detailed insight into the nature of specific reactions, but benchmark results on which to base more approximate approaches, such as transition state theory and quasiclassical trajectories, which can be applied to larger systems. 

Figure 22 shows an application of the present method to the H3 reaction system and the thermal rate constant is calculated. The final result with tunneling effects included agree well with the quantum mechanical transition state theory calculations, although the latter is not shown here. 

Kinetics on the level of individual molecules is often referred to as reaction dynamics. Subtle details are taken into account, such as the effect of the orientation of molecules in a collision that may result in a reaction, and the distribution of energy over a molecule s various degrees of freedom. This is the fundamental level of study needed if we want to link reactivity to quantum mechanics, which is really what rules the game at this fundamental level. This is the domain of molecular beam experiments, laser spectroscopy, ah initio theoretical chemistry and transition state theory. It is at this level that we can learn what determines whether a chemical reaction is feasible. 

Transition state theory, as embodied in Eq. 10.3, or implicitly in Arrhenius theory, is inherently semiclassical. Quantum mechanics plays a role only in consideration of the quantized nature of molecular vibrations, etc., in a statistical fashion. But, a critical assumption is that only those molecules with energies exceeding that of the transition state barrier may undergo reaction. In reality, however, the quantum nature of the nuclei themselves permits reaction by some fraction of molecules possessing less than the energy required to surmount the barrier. This phenomenon forms the basis for QMT. ... 

Beyond Transition State Theory (and, therefore, beyond Monte Carlo simulations) dynamical effects coming from recrossings should be introduced. Furthermore, additional quantum mechanical aspects, like tunneling, should be taken into account in some chemical reactions. 

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