Other Variational Transition State Theories

Canonical variational theory finds the best dividing surface for a canonical ensemble, characterized by temperature T, to minimize the calculated canonical rate constant. Alternative variational transition state theories can also be 

The location of the dividing surface that minimizes Eq. [22] is defined as which specifies the microcanonical variational transition state thus, 

Notice that the minimum-number-of-states criterion corresponds correctly to variational transition state theory, whereas an earlier minimum-density-of-states criterion does not. The microcanonical rate constant can be written as 

As we go to the right in the above sequence, the methods accoiuit more accurately for recrossing effects. 

Sometimes it is found that even the best dividing surface gives too high rate constants because another reaction bottleneck exists. Those cases can be handled, at least approximately, by the luiified statistical (US) model. In this method, the thermal rate constant can be written as 

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Calculations have identified three transition states (TS) for an SN2 reaction.4"6 Two are variational, one of which is located along the X + RY association reaction path, and the other along the XR + Y" association reaction path i.e. see Figure 1. Variational transition state theory (VTST) calculations show that the third TS is located at the central barrier.4... 

Free energy is the key quantity that is required to determine the rate of a chemical reaction. Within the Conventional Transition State Theory, the rate constant depends on the free energy barrier imposed by the conventional transition state. On the other hand, in the frame of the Variational Transition State Theory, the free energy is the magnitude that allows the location of the variational transition state. Then, it is clear that the evaluation of the free energy is a cornerstone (and an important challenge) in the simulation of the chemical reactions in solution... 

To begin we are reminded that the basic theory of kinetic isotope effects (see Chapter 4) is based on the transition state model of reaction kinetics developed in the 1930s by Polanyi, Eyring and others. In spite of its many successes, however, modern theoretical approaches have shown that simple TST is inadequate for the proper description of reaction kinetics and KIE s. In this chapter we describe a more sophisticated approach known as variational transition state theory (VTST). Before continuing it should be pointed out that it is customary in publications in this area to use an assortment of alphabetical symbols (e.g. TST and VTST) as a short hand tool of notation for various theoretical methodologies. 

Of course, one is not really interested in classical mechanical calculations. Thus in normal practice the partition functions used in TST, as discussed in Chapter 4, are evaluated using quantum partition functions for harmonic frequencies (extension to anharmonicity is straightforward). On the other hand rotations and translations are handled classically both in TST and in VTST, which is a standard approximation except at very low temperatures. Later, by introducing canonical partition functions one can direct the discussion towards canonical variational transition state theory (CVTST) where the statistical mechanics involves ensembles defined in terms of temperature and volume. There is also a form of variational transition state theory based on microcanonical ensembles referred to by the symbol p,. Discussion of VTST based on microcanonical ensembles pVTST is beyond the scope of the discussion here. It is only mentioned that in pVTST the dividing surface is... 

The second approach, a multidimensional one, was given by Langer [7], Other multidimensional developments were many [16-18]. McCammon [17] discussed a variational approach (1983) to seek the best path for crossing the transition-state hypersurface in multidimensional space and discussed the topic of saddle-point avoidance. Further developments have been made using variational transition state theory, for example, by Poliak [18]. 

Variational transition state theory was suggested by Keck [36] and developed by Truhlar and others [37,38]. Although this method was originally applied to canonical transition state theory, for which there is a unique optimal transition state, it can be applied in a much more detailed way to RRKM theory, in which the transition state can be separately optimized for each energy and angular momentum [37,39,40]. This form of variational microcanonical transition state theory is discussed at length in Chapter 2, where there is also a discussion of the variational optimization of the reaction coordinate. 

The RBU model can be used to study the effect of exciting the vibrational modes treated within the model. For the reactions X (X=C1, 0 and H) + CH4 HX + CH3 we find that exciting a vibrational inode results in a lower threshold to reaction. It was also found that exciting the reactive C-H stretch enhances the reactivity more than exciting the CH4 umbrella mode. Vibrational enhancements for the umbrella and C-H stretch vibrations have also been found in other studies of the dynamics[75, 80] and in canonical variational transition state theory (C T) calculations [84]. Enhancement of the Cl + CH4 reaction due to vibrational excitation of the H-CH3 stretch has also been confirmed b experimental measurements by Zare and coworkers[85]. 

Warshel and Chu [42] and Hwang et al. [60] were the first to calculate the contribution of tunneling and other nuclear quantum effects to PT in solution and enzyme catalysis, respectively. Since then, and in particular in the past few years, there has been a significant increase in simulations of quantum mechanical-nuclear effects in enzyme and in solution reactions [16]. The approaches used range from the quantized classical path (QCP) (for example. Refs. [4, 58, 95]), the centroid path integral approach [54, 55], and variational transition state theory [96], to the molecular dynamics with quantum transition (MDQT) surface hopping method [31] and density matrix evolution [97-99]. Most studies of enzymatic reactions did not yet examine the reference water reaction, and thus could only evaluate the quantum mechanical contribution to the enzyme rate constant, rather than the corresponding catalytic effect. However, studies that explored the actual catalytic contributions (for example. Refs. [4, 58, 95]) concluded that the quantum mechanical contributions are similar for the reaction in the enzyme and in solution, and thus, do not contribute to catalysis. 

T. D. (2004) Ensemble-averaged variational transition state theory with optimized multidimensional tunneling for enzyme kinetics and other condensed-phase reactions,... 

In this contribution, we present a preliminary account of our recent work on the Cl + H2 and Cl + D2 reactions, which includes a new potential energy surface, variational transition state theory and semiclassical tunneling calculations for both reactions and for other isotopomeric cases, and accurate quantum dynamical calculations of rate constants and state-to-state integral and differential cross sections. 

Tmhlar DG, Gao JL, Garcia-Viloca M, Alhambra C, Corchado J, Sanchez ML, Poulsen TD (2004) Ensemble-averaged variational transition state theory with optimized multidimensional tunneling for enzyme kinetics and other condensed-phase reactions, hit J Quantum Chem 100(6) 1136-1152... 

The double hydrogen bond in principle facilitates two mechanisms of double proton transfer stepwise and concerted. Studying the transfer rates within the semi-classical tunneling approximation of the variational transition state theory, Kim found that the two protons are transferred synchronously across the transition state with D2h symmetry [194], The actual proton-tunneling distance is considerably reduced due to the contraction of the hydrogen bonds, i.e., heavy atom motion promotes the proton transfer. On the other hand, the ab initio path-integral Car-Parinello calculations predict that the motion of the two protons in the vicinity of the potential minima is... 

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