Transition states nonequilibrium solvation

A new issue arises when one makes a solute-solvent separation. If the solvent enters the theory only in that V(R) is replaced by TT(R), the treatment is called equilibrium solvation. In such a treatment only the coordinates in the set R can enter into the definition of the transition state. This limits the quality of the dynamical bottleneck that one can define depending on the system, this limitation may cause small quantitative errors or larger more qualitative ones, even possibly missing the most essential part of a reaction coordinate (in a solvent-driven reaction). Going beyond the equilibrium solvation approximation is called nonequilibrium solvation or solvent friction [4,26-28], This is discussed further in Section 3.3.2. 

The most useful theoretical framework for studying chemical reactions in solution is transition state theory. Building on the material presented in the introduction, we will begin by presenting a general theory called the equilibrium solvation path (ESP) theory of reactions in a liquid. We then present an approximation to ESP theory called separable equilibrium solvation (SES). Finally we present a more complete theory, still based on an implicit treatment of solvent, called nonequilibrium solvation (NES). All three... 

The SES, ESP, and NES methods are particularly well suited for use with continuum solvation models, but NES is not the only way to include nonequilibrium solvation. A method that has been very useful for enzyme kinetics with explicit solvent representations is ensemble-averaged variational transition state theory [26,27,87] (EA-VTST). In this method one divides the system into a primary subsystem and a secondary one. For an ensemble of configurations of the secondary subsystem, one calculates the MEP of the primary subsystem. Thus the reaction coordinate determined by the MEP depends on the coordinates of the secondary subsystem, and in this way the secondary subsystem participates in the reaction coordinate. 

In terms of traditional Transition State Theory (TST) for solution reactions [40,41], in which e.g. the activation free energy AG can be estimated with equilibrium solvation dielectric continuum theories [42-46], the nonequilibrium or dynamical solvation aspects enter the prefactor of the rate constant k, or in terms of the ratio of k to its TST approximation kTST, k, the transmission coefficient, k and kTST are related by [41]... 

We have already mentioned in the Introduction (Section 3.7.1) the importance of conical intersections (CIs) in connection with excited electronic state dynamics of a photoexcited chromophore. Briefly, CIs act as photochemical funnels in the passage from the first excited S, state to the ground electronic state S0, allowing often ultrafast transition dynamics for this process. (They can also be involved in transitions between excited electronic states, not discussed here.) While most theoretical studies have focused on CIs for a chromophore in the gas phase (for a representative selection, see refs [16, 83-89], here our focus is on the influence of a condensed phase environment [4-9], In particular, as discussed below, there are important nonequilibrium solvation effects due to the lack of solvent polarization equilibration to the evolving charge distribution of the chromophore. 

Such a splitting in the medium response gives rise to the so-called nonequilibrium solvation regime. In the case of a vertical electronic transition (from the GS to an excited state for absorption, or from an excited state to the GS for emission), the arrival state feels a nonequilibrium solvation regime as the characteristic time of the electronic transition is much shorter than the response time of the inertial components of the solvent, and this component remains equilibrated with the initial electronic state. The arrival state reaches an equilibrium solvation regime only if its life time is enough to allow for a complete relaxation of the slow (inertial) polarization of the solvent. 

We have seen that dynamical solvent effects in the friction can lead to a breakdown of TST. As stressed above, this is also a breakdown in the equilibrium solvation assumption for the transition state and configurations in its neighborhood. In fact, the standard TST view is a special one-dimensional equilibrium perspective, i.e. a mean potential curve for the reacting species is visualized and no friction of any sort is considered. The solvent influence can be felt solely via this potential, hence it is assumed that for each configuration of the reacting species, the solvent is equilibrated. On the contrary, the discussion above about Kramers and Grote-Hynes theories has documented the importance of nonequilibrium solvation effects in a frictional language. 

Where k is the transmission factor, < x >xs is the average of the absolute value of the velocity along the reaction coordinate at the transition state (TS), and P = l/keT ( vhere ke is the Boltzmann constant and T the absolute temperature). The term AG designates the multidimensional activation free energy that expresses the probability that the system vill be in the TS region. The free energy reflects enthalpic and entropic contributions and also includes nonequilibrium solvation effects [4] and, as will be shown below, nuclear quantum mechanical effects. It is also useful to comment here on the common description of the rate constant as... 

S. C. Tucker and D. G. Truhlar, /. Am. Chem. Soc., 112, 3347 (1990). The Effect of Nonequilibrium Solvation on Chemical Reaction Rates. Variational Transition State Theory Studies of the Microsolvated Reaction C1 (H20) + CH3CI. 

D. G. Truhlar, G. K. Schenter, and B. C. Garrett,/. Chem. Phys., 98,5756 (1993). Inclusion of Nonequilibrium Continuum Solvation Effects in Variation Transition State TTieory. 

We will first give an overview of the issues involved via a brief description of the Transition State Theory and the dynamic Grote-Hynes Theory, as developed for charge transfer reactions in solution by van der Zwan and Hynes.This will introduce the ideas of equilibrium and nonequilibrium solvation, friction and barrier recrossing. We then indicate some of the consequences and predictions for the Sfjl and Sfj2 reaction types. 

Finally, we consider a condensed phase example, namely ci -FCH3 C1 CH3 CH- Cr in aqueous solution. Why does this reaction have a 26 kcal mol activation energy in solution, when in the gas-phase the barrier to interconversion has a lower energy than reactants The answer is that the equilibrium free energy of solvation of the transition state is 33 kcal mol less negative than the free energy of solvation of reactants Nonequilibrium effects are smaller but have been variously estimated to lower the reaction rates a further 4-57%. " ... 

In this contribution we have presented some specific aspects of the quantum mechanical modelling of electronic transitions in solvated systems. In particular, attention has been focused on the ASC continuum models as in the last years they have become the most popular approach to include solvent effects in QM studies of absorption and emission phenomena. The main issues concerning these kinds of calculations, namely nonequilibrium effects and state-specific versus linear response formulations, have been presented and discussed within the most recent developments of modern continuum models. 

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