Tunnel effect theory rate constant calculations

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. 

CVT approach is particularly attractive due to the limited amount of potential energy and Hessian information that is required to perform the calculations. Direct dynamics with CVT thus offers an efficient and cost-effective methodology. Furthermore, several theoretical reviews60,61 have indicated that CVT plus multidimensional semi-classical tunneling approximations yield accurate rate constants not only for gas-phase reactions but also for chemisorption and diffusion on metals. Computationally, it is expensive if these Hessians are to be calculated at an accurate level of ab initio molecular orbital theory. Several approaches have been proposed to reduce this computational demand. One approach is to estimate rate constants and tunneling contributions by using Interpolated CVT when the available accurate ab initio electronic structure information is very limited.62 Another way is to carry out CVT calculations with multidimensional semi-classical tunneling approximations. 

Inclusion of dynamical effects allows calculation of corrections to simple Transition State Theory, often described by a transmission coefficient k to be multiplied with the TST rate constant (Section 12.1), or used in connection with variational TST (Section 12.3). Classical dynamics allow corrections due to recrossing to be calculated, while a quantum treatment is necessary for including tunnelling effects. Owing to the stringent... 

A recent tunneling effect model for radiationless transitions (198) has been applied to benzene and other aromatic hydrocarbons. The CH stretching vibrations are considered as dominant for the non-radiatlve process. Rate constants for the radiationless process Sg calculated by theory are of the same order of magni-... 

Hwang et al.131 were the first to calculate the contribution of tunneling and other nuclear quantum effects to enzyme catalysis. Since then, and in particular in the past few years, there has been a significant increase in simulations of QM-nuclear effects in enzyme reactions. The approaches used range from the quantized classical path (QCP) (e.g., Refs. 4,57,136), the centroid path integral approach,137,138 and vibrational TS theory,139 to the molecular dynamics with quantum transition (MDQT) surface hopping method.140 Most studies did not yet examine the reference water reaction, and thus could only evaluate the QM contribution to the enzyme rate constant, rather than the corresponding catalytic effect. However, studies that explored the actual catalytic contributions (e.g., Refs. 4,57,136) concluded that the QM contributions are similar for the reaction in the enzyme and in solution, and thus, do not contribute to catalysis. 

Various quantum-mechanical theories have been proposed which allow one to calculate isotopic Arrhenius curves from first principles, where tunneling is included. These theories generally start with an ab initio calculation of the reaction surface and use either quantum or statistical rate theories in order to calculate rate constants and kinetic isotope effects. Among these are the variational transition state theory of Truhlar [15], the instanton approach of Smedarchina et al. 

In general, proton transfer occurs via a combination of over-barrier and through-barrier pathways. The rate constant of over-barrier transfer is usually calculated by standard transition state theory (TST) [22] by separating the reaction coordinate from the remaining degrees of freedom. If tunneling effects and the curvature of the reaction path are neglected, this leads to the expression... 

Joseph, T.R., Steckler, R. and Truhlar, D.G. (1987) A new potential energy surface for the CH3 + H2 CH + H reaction Calibration and calculation of rate constants and kinetic isotope effects by variational transition state theory and semi-classical tunneling calculations, J. Chem. Phys. 87, 7036-7049. 

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. 

Where k is the rate constant, k is the transmission coefficient from the collision theory, Piunn is the tunneling correction, and the Qs are the standard partition functions (see Felipe et al. in this volume). By substituting appropriate carbon atoms with different isotopes on the molecule, one can calculate the ratio of the rate constants (kinetics isotopic effect). 

An overall scheme for quantitative assessment of the influence of tunnelling effects upon the reaction rate has been developed by Miller [113, 114]. The simplest method for calculating the tunnelling rate constant is based on the theory of transition state with correction for tunnelling. This correction consists in formal replacement of the classical motion along the reaction coordinate with the quantum motion. This approach was first formulated in the works by Bell [115]. 

One fundamental assumption in classical transition state theory is that of no recrossing over the transition state. The advantage, of the RP theory is therefore that since it provides a hamiltonian it can be used in dynamical calculations thereby incorporating the effect of recrossing. However, it is also possible to use the hamiltonian for an estimate of the transmission factor, i.e. the correction to transition state theory from recrossing of the trajectories. An additional correction factor comes from quantum tunneling (see below). Considering the reaction rate constant it may be expressed as... 

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