Variational transition-state theory tunneling corrections

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] ... 

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). 

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. 

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. 

Computational methods now exist that include contributions from all vibrational modes to the H/D-transfer process, thus eliminating the need to introduce any empirical parameters, e.g., variational transition state theory with semiclassical tunneling corrections (Truhlar, D. G. Garett, B. C. Klippen-stein, S. J.J. Phys. Chem. 1996, 100, 12771) and the approximate instanton method (Siebrand, W Smedarchina, Z. Zgierski, M. Z. Femandez-Ramos, A. Int. Rev. Chem. Phys. 1999, 18, 5). 

A more sophisticated reaction path approach is to replace in eq. (40) by 8jjq(sJj This is the essence of the approach taken by Garrett and Truhlar and generalized by Miller et al. and Skodje and Truhlar for polyatomic reactions. Truhlar and coworkers have proposed one-dimensional paths which deviate from the reaction path in order to compute accurate tunneling probabilities from which transmission coefficients (see below) are then used to correct their version of variational transition state theory, the so-called improved canonical variational [transition state] theory (ICVT) (also see below). 

In this Section we discuss and compare the results obtained using the various theories of Section 3, and a variational transition state theory with a tunnelling correction [9]. The reaction we concentrate on is H+BrH - HBr+H. We emphasize the rate constants, but also discuss reaction cross sections. The potential energy surface we used in the H+BrH computations is of a semiempirical diatomics in molecules type, the form of which (called DIM-3C) is due to Last and Baer [ll]. The surface contains a three-centre integral term that has been parameter-ised [33] by comparing ESA-CSA calculations with an experimental [3] room temperature rate constant for the D+BrH -) DBr+H reaction. The minimum potential energy path is collinear, and there is a strong... 

If information on the reaction path is available, as, for instance, in variational transition state theory, this can be used to calculate k [69,70]. In transition state theory, only the knowledge of the energy and its first and second derivatives at the reactant and transition state locations is needed and the barrier is typically approximated by a simple functional form. One possibility is to describe the reaction barrier by an Eckart potential [75] (also called a sech potential, depending on the literature source), k in Eq. (7.19) is defined as the ratio of transmitted quantum particles to classical particles and the resulting integral for the Eckart potential can be solved numerically. An approximate solution is the Wigner tunneling correction ... 

The AG value deduced from the PMF is corrected by replacing classical vibrational partition functions by their quantum homolog. Recrossing, tunnelling and non-classical reflection effects can be included in the transmission coefficient by various procedures. This ensemble-average variational transition state theory with multidimensional tunnelling (EA-VTST/MT) method was applied to proton and hydride transfers in various enzymes such as yeast enolase, liver alcohol dehydrogenase and triosephosphate isomerase. For a review, see ref. 3 and the chapter by J. Gao in this book. 

The results discussed in the previous section reveal that the reaction rate corresponding to the formation of major by-products of the oxidation reaction is important to determine the lifetime of dimethylphenol in the atmosphere. The rate constants are calculated using canonical variational transition state theory (CVT) with small curvature tunneling (SCT) corrections over the temperature range of 278-350 K. As described in Figure 19.2, the formation of product channels consists of four reaction channels. The rate constants for the formation of alkyl radical (11), peroxy radical (12), m-cresol and the product channels are designated as k, ki2, and kp, respectively, and are summarized in Tables 19.2 and 19.3. The reaction path properties and rate constant obtained for the most favorable product channels, P5 and P6, are discussed in detail. 

While the MRCI(322)- -Q-corrected CCI4-Q PES should be accurate, direct comparison with experiment is difficult. To facilitate comparison we have employed canonical variation transition state theory [61] at the classical and adiabatic barrier using the CCI-I-Q potential for both F-I-H2 and F-I-D2. These calculations account for the zero-point energy and include a tunneling correction. The results of these calculations Me summarized in Table IX. As expected, the zero-point and tunneling corrections Me different for H2 and D2. At the classical saddle-point. 

In Fig. 5.1, thermal rate constants obtained from accurate full-dimensional calculations are compared to transition state theory results [40] (left panel) and reduced dimensionality quantum calculations (right panel). Experimental results [41] are also displayed. Classical transition state theory (TST) and variational transition state theory (VTST) drastically underestimate the thermal rate constant. This shows that tunneling effects are very prominent in the H2 + OH reaction. Including different types of tunneling corrections [40], increased values of the thermal rate constant can be obtained. However, it should be noted that different tunneling corrections result in considerably different results. This reflects the ambiguity of quantum transition state... 

Variational Transition-State Theory and Semiclassical Tunneling Calculations With Interpolated Corrections A New Approach to Interfacing Electronic Stmcture Theory and... 

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... 

Calculations using transition state theory (TST) are the subject of an other article (see Transition State Theory). This method gives rate constants for chemical reactions, but cannot normally give the energy resolved or quantum state-to-state detail that is needed for comparison with, for example, the results of molecular beam experiments. Sophisticated versions of transition state theory (that include, for example, variational placement of the transition state, optimum reaction paths for particular mass combinations, and tunneling corrections) have been applied to several reactions including those involving polyatomic molecules. Examples include ... 

---