Quantum instanton approximation

In chemical kinetics, one can often learn much by knowing only the kinetic isotope effect or the temperature dependence of the rate constant, instead of the rate constant itself. As various systematic errors can be avoided, the temperature dependence and kinetic isotope effect are easier to measure experimentally. Similarly, this chapter described a theoretical formalism based on the quantum instanton approximation to compute these two properties directly, without computing the rate constant itself. The common ingredients are the thermodynamic integration with respect to a specific parameter (inverse temperature or the isotope mass), path integral implementation, and, last but not least, the use of efficient path integral estimators. 

While the quantum instanton approximation cannot describe the recrossing effect, this effect, which is predominantly classical, could be included empirically as a multiplicative factor (a transmission coefficient ) obtained in a classical molecular dynamics simulation. The main challenge for future research is to develop accurate and efficient approximations for the rate constant, its temperature dependence, and the kinetic isotope effect that include the quantum and recrossing effects simultaneously and rigorously. 

Methods used Arrhenius law, transition state theory, Wigner tunneling correction, quantum instanton approximation, and exact scattering calculation... 

Methods used transition state theory and quantum instanton approximation... 

Zhao, Y., Yamamoto, T., and Miller, W.H. (2004) Path integral calculation of thermal rate constants within the quantum instanton approximation application to the h + chjsub 4] -> hjsub 2] + chjsub 3] hydrogen abstraction reaction in full cartesian space. J. Chem. Phys., 120 (7), 3100-3107. 

Buchowiecki, M. and Vamdek, J. (2010) Direct evaluation of the temperature dependence of the rate constant based on the quantum instanton approximation. /. Chem. Phys., 132, 194106. 

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]. 

Taking again MA as an example, we determine the primary H/D KIE for the intramolecular PT in MA at various temperatures by combining the MMPT force field with path-integral Monte Carlo (PIMC) simulations [106]. Employing the quantum instanton (QI) approximation [107], The KIE is expressed as... 

It should be noted that in the cases where y"j[,q ) > 0, the centroid variable becomes irrelevant to the quantum activated dynamics as defined by (A3.8.Id) and the instanton approach [37] to evaluate based on the steepest descent approximation to the path integral becomes the approach one may take. Alternatively, one may seek a more generalized saddle point coordinate about which to evaluate A3.8.14. This approach has also been used to provide a unified solution for the thennal rate constant in systems influenced by non-adiabatic effects, i.e. to bridge the adiabatic and non-adiabatic (Golden Rule) limits of such reactions. 

---