Tunneling variational transition states

Many computational studies in heterocyclic chemistry deal with proton transfer reactions between different tautomeric structures. Activation energies of these reactions obtained from quantum chemical calculations need further corrections, since tunneling effects may lower the effective barriers considerably. These effects can either be estimated by simple models or computed more precisely via the determination of the transmission coefficients within the framework of variational transition state calculations [92CPC235, 93JA2408]. 

The rate of hydrogen transfer can be calculated using the direct dynamics approach of Truhlar and co-workers which combines canonical variational transition state theory (CVT) [82, 83] with semi-classical multidimensional tunnelling corrections [84], The rate constant is calculated using [83] ... 

Allison TC, Trahlar DG (1998) Testing the accuracy of practical semiclassical methods variational transition state theory with optimized multidimensional tunnelling. In Thompson DL (ed) Modern Methods for Multidimensional Dynamics Computations in Chemistry. World Scientific, Singapore, p 618... 

Fernandez-Ramos A, Ellingson BA, Garrett BC, Trahlar DG (2007) Variational transition state theory with multidimensional tunneling. In Lipkowitz KB, Cundari TR, Boyd DB (eds) Reviews in Computational Chemistry, Vol 23. Wiley-VCH, New York, p 125... 

Antonio Fernandez-Ramos, Benjamin A. Ellingson, Bruce C. Garrett, and Donald G. Truhlar, Variational Transition State Theory with Multidimensional Tunneling. 

Kinetic Isotope Effects Continued Variational Transition State Theory and Tunneling... 

Abstract Some of the successes and several of the inadequacies of transition state theory (TST) as applied to kinetic isotope effects are briefly discussed. Corrections for quantum mechanical tunneling are introduced. The bulk of the chapter, however, deals with the more sophisticated approach known as variational transition state theory (VTST). 

Tests of Variational Transition State Theory (Including Tunneling)... 

Table 6.2 Tests of Variational Transition State Theory by Comparing with Exact Quantum Calculations (Extracted from Allison, T. C. and Truhlar, D. G. Testing the accuracy of practical semiclassical methods variational transition state theory with optimized multidimensional tunneling, in Thompson, D. L., Ed. Modem methods for multidimensional dynamics computations in chemistry, World Scientific, Singapore 1998. pp 618-712. This reference quotes results on many more reactions and BO surfaces over broad temperature ranges.)The numbers in the table are ratios of the results of the approximate calculation to the quantum calculation, all at 300 K...

TST = conventional Transition State Theory, ICVT = Improved Canonical Variational Transition state theory, ICVT/SCT = ICVT/Small Curvature Tunneling, ICVT/p,OMT = ICVT/Microcanonical Optimized Multidimensional Tunneling. 

Truhlar, D. G. Variational transition state theory and multidimensional tunneling for simple and complex reactions in the gas phase, solids, liquids, and enzymes, in Kohen, A. and Limbach, H. H., Eds. Isotope Effects in Chemistry and Biology. CRC Press/Taylor Francis, Boca Raton, FL (2006), pp. 579-619. 

Alhambra and co-workers adopted a QM/MM strategy to better understand quantum mechanical effects, and particularly the influence of tunneling, on the observed primary kinetic isotope effect of 3.3 in this system (that is, the reaction proceeds 3.3 times more slowly when the hydrogen isotope at C-2 is deuterium instead of protium). In order to carry out their analysis they combined fully classical MD trajectories with QM/MM modeling and analysis using variational transition-state theory. Kinetic isotope effects (KIEs), tunneling, and variational transition state theory are discussed in detail in Chapter 15 - we will not explore these topics in any particular depth in this case study, but will focus primarily on the QM/MM protocol. 

Tucker, S. C. and Truhlar, D. G. 1989. Dynamical Formulation of Transition State Theory Variational Transition States and Semiclassical Tunneling , in New Theoretical Concepts for Understanding Organic Reactions, Bertran, J. and Czismadia, I. G., Eds., Kluwer Berlin, 291. 

In ESP theory [30-32] we treat the system by the same methods that we would use in the gas phase except that in the nontunneling part of the calculation we replace V(R) by TT(R), and in the tunneling part we approximate V(R) by TT(R) or a function of TT(R). Next we review what that entails. In particular we will review the application of variational transition state theory [21-25] with optimized multidimensional tunneling [33,34] to liquid-phase reactions for the case [31,32] in which TT(R) is calculated from V(R) by... 

T. V. Albu, J. C. Corchado, D. G. Truhlar, J. Phys. Chem. A 105, 8465 (2001). Molecular Mechanics for Chemical Reactions A Standard Strategy for Using Multiconfiguration Molecular Mechanics for Variational Transition State Theory with Optimized Multidimensional Tunneling. 

Theoretical studies of the microsolvation effect on SN2 reactions have also been reported by our coworkers and ourselves (Gonzalez-Lafont et al. 1991 Truhlar et al. 1992 Tucker and Truhlar 1990 Zhao et al. 1991b, 1992). Two approaches were used for interfacing electronic structure calculations with variational transitional state theory (VST) and tunneling calculations. We analyzed both the detailed dynamics of microsolvation and also its macroscopic consequences (rate coefficient values and kinetic isotope effects and their temperature... 

Both approaches include tunneling corrections and provide approximately the same accuracy. However, the variational transition state theory is computationally quite demanding, and at least 40 points on the path of the proton transfer should be available. In contrast, the instanton approach uses only vibrational frequencies calculated for local minima and transition states and corresponding values of energy. 

Gonzalez-Lafont A, Troung TN, Truhlar DG (1991) Interpolated variational transition-state theory practical methods for estimating variational transition-state properties and tunneling contributions to chemical reaction rates from electronic structure calculations, J Phys Chem 95 8875-8894... 

Quantum dynamics effects for hydride transfer in enzyme catalysis have been analyzed by Alhambra et. al., 2000. This process is simulated using canonically variational transition-states for overbarrier dynamics and optimized multidimensional paths for tunneling. A system is divided into a primary zone (substrate-enzyme-coenzyme), which is embedded in a secondary zone (substrate-enzyme-coenzyme-solvent). The potential energy surface of the first zone is treated by quantum mechanical electronic structure methods, and protein, coenzyme, and solvent atoms by molecular mechanical force fields. The theory allows the calculation of Schaad-Swain exponents for primary (aprim) and secondary (asec) KIE... 

There are two corrections to equation (12) that one might want to make. The first has to do with dynamical factors [19,20] i.e., trajectories leave Ra, crossing the surface 5/3, but then immediately return to Ra. Such a trajectory contributes to the transition probability Wfia, but is not really a reaction. We can correct for this as in variational transition-state theory (VTST) by shifting Sajj along the surface normals. [8,9] The second correction is for some quantum effects. Equation (14) indicates one way to include them. We can simply replace the classical partition functions by their quantum mechanical counterparts. This does not correct for tunneling and interference effects, however. 

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