Detailed quantum transition state theory

Quantum mechanical effects—tunneling and interference, resonances, and electronic nonadiabaticity— play important roles in many chemical reactions. Rigorous quantum dynamics studies, that is, numerically accurate solutions of either the time-independent or time-dependent Schrodinger equations, provide the most correct and detailed description of a chemical reaction. While hmited to relatively small numbers of atoms by the standards of ordinary chemistry, numerically accurate quantum dynamics provides not only detailed insight into the nature of specific reactions, but benchmark results on which to base more approximate approaches, such as transition state theory and quasiclassical trajectories, which can be applied to larger systems. 

Kinetics on the level of individual molecules is often referred to as reaction dynamics. Subtle details are taken into account, such as the effect of the orientation of molecules in a collision that may result in a reaction, and the distribution of energy over a molecule s various degrees of freedom. This is the fundamental level of study needed if we want to link reactivity to quantum mechanics, which is really what rules the game at this fundamental level. This is the domain of molecular beam experiments, laser spectroscopy, ah initio theoretical chemistry and transition state theory. It is at this level that we can learn what determines whether a chemical reaction is feasible. 

Quantitative estimates of E are obtained the same way as for the collision theory, from measurements, or from quantum mechanical calculations, or by comparison with known systems. Quantitative estimates of the A factor require the use of statistical mechanics, the subject that provides the link between thermodynamic properties, such as heat capacities and entropy, and molecular properties (bond lengths, vibrational frequencies, etc.). The transition state theory was originally formulated using statistical mechanics. The following treatment of this advanced subject indicates how such estimates of rate constants are made. For more detailed discussion, see Steinfeld et al. (1989). 

The theoretical aspects of electron transfer mechanisms in aqueous solution have received considerable attention in the last two decades. The early successes of Marcus Q, 2), Hush (3, 4), and Levich (5) have stimulated the development of a wide variety of more detailed models, including those based on simple transition state theory, as well as more elaborate semi-clas-sical and quantum mechanical models (6-12). 

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. 

The statistical theories provide a relatively simple model of chemical reactions, as they bypass the complicated problem of detailed single-particle and quantum mechanical dynamics by introducing probabilistic assumptions. Their applicability is, however, connected with the collisional mechanism of the process in question, too. The statistical phase space theories, associated mostly with the work of Light (in Ref. 6) and Nikitin (see Ref. 17), contain the assumption of a long-lived complex formation and are thus best suited for the description of complex-mode processes. On the other hand, direct character of the process is an implicit dynamical assumption of the transition-state theory. 

As the fundamental concepts of chemical kinetics developed, there was a strong interest in studying chemical reactions in the gas phase. At low pressures the reacting molecules in a gaseous solution are far from one another, and the theoretical description of equilibrium thermodynamic properties was well developed. Thus, the kinetic theory of gases and collision processes was applied first to construct a model for chemical reaction kinetics. This was followed by transition state theory and a more detailed understanding of elementary reactions on the basis of quantum mechanics. Eventually, these concepts were applied to reactions in liquid solutions with consideration of the role of the non-reacting medium, that is, the solvent. 

Especially in the early stages of mechanism development, it is not likely to be practical to carry out quantum chemistry and transition state theory calculations for all the possible reactions in the mechanism. So, simpler empirical methods must be used to obtain initial estimates of the rate parameters. Perhaps the most useful description of these methods is found in the classic text. Thermochemical Kinetics, by Benson [42]. The following rules of thumb for estimating rate parameters closely follow what is presented in more detail there. For different classes of reactions, the framework of transition state theory can be applied to aid in the estimation of rate parameters. This is most... 

For chemical kinetics, transition state theory is most useM in the form that starts from reactants in thermal equilibrium. For our purpose we want a more detailed version, that of reactants with a total energy in the range E oE + AE. If we know how to do that, we can and will average over a Boltzmann distribution in E to obtain the thermal results. The first task at hand is to define what is meant by reactants at equilibrium at a total energy within the range (and at given values of any other conserved quantum numbers). It is the foundation of statistical mechanics that equilibrium under such conditions means that all possible quantum states of the reactants are equally probable. ... 

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 ... 

The sole factor determining the value of a rate or equilibrium constant is the difference in free energy between the reactants and either a transition state, in the case of a reaction rate, or the products, in the case of an equilibrium. Rate and equilibrium constants cannot therefore properly be correlated with any static property of the reactants themselves. This point cannot be emphasized too strongly in view of the many attempts that have been made to find such correlations. One might add that attempts of this kind are in any case basically unsatisfactory in that they ignore the transition state entirely and so can throw no light on its structure even if correlations of this kind can be established, they do not throw any light on the detailed mechanism of chemical process—and it is just in this field that quantum theory has most to offer to chemists. 

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