Nonequilibrium Solvation Path

The SES and ESP approximations include the dynamics of solute degrees of freedom as fully as they would be treated in a gas-phase reaction, but these approximations do not address the full complexity of condensed-phase reactions because they do not allow the solvent to participate in the reaction coordinate. Methods that allow this are said to include nonequilibrium solvation. A variety of ways to include nonequilibrium solvation within the context of an implicit or reduced-degree-of-freedom bath are reviewed elsewhere [69]. Here we simply discuss one very general such NES method [76-78] based on collective solvent coordinates [71, 79]. In this method one replaces the solvent with one or more collective solvent coordinates, whose parameters are fit to bulk solvent properties or molecular dynamics simulations. Then one carries out calculations just as in the gas phase but with these extra one or more degrees of freedom. The advantage of this approach is its simplicity (although there are a few subtle technical details). 

A difficulty with the nonequilibrium approach is that one must estimate the time constant or time constants for solvent equilibration with the solvent. This may be estimated from solvent viscosities, from diffusion constants, or from classical trajectory calculations with explicit solvent. Estimating the time constant for solvation dynamics presents new issues because there is more than one relevant time scale [69, 80]. Fortunately, though, the solvation relaxation time seems to depend mostly on the solvent, not the solute. Thus it is very reasonable to assume it is a constant along the reaction path. 

Another difficulty with the NES model is not knowing how reliable the solvent model is and having no systematic way to improve it to convergence. Furthermore this model, like the SES and ESP approximations, assumes that the reaction can be described in terms of a reaction path residing in a single free energy valley or at most a small number of such valleys. The methods discussed next are designed to avoid that assumption. 

The ESP method was applied to the reaction mentioned in Subsection 27.4.2, namely 1,2-hydrogen migration in chloromethylcarbene. Tunneling contributions are found to be smaller in solution than in the gas phase, but solvation by 1,2-di-chloroethane lowers the Arrhenius activation energy at 298 K from 7.7 kcal mol i to 6.0 kcal mol i [67[. 

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Even at this level of dynamical theory, one is not restricted to considering equilibrium solvation of the gas-phase saddle point or of configurations along the gas-phase reaction path [109, 338-344], and to the extent that the solvent is allowed to affect the choice of the reaction path itself, dynamic (i.e., nonequilibrium) solvation effects begin to appear in the theory. 

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... 

Of course, there is more to a chemical reaction than its rate constant the reaction path or mechanism is also of central interest. Once again, nonequilibrium solvation is crucial in describing this path. In an equilibrium solvation picture, the solvent polarization would remain equilibrated throughout the reaction course, but this assumption is rarely satisfied for an actual reaction path, because of the same considerations noted above for the rate constant. Indeed these nonequilibrium solvation effects can qualitatively change the character of the reaction path as compared with an equilibrium solvation image. Dielectric continuum dynamic descriptions thus have an important role to play here as well. Indeed, we will employ in this contribution the reaction path Hamiltonian formulation previously developed [48,49], which can be used to generate a reaction path which is the analog in solution of the well-known Fukui reaction path in the gas phase [50], The reaction path will be discussed for both reaction topics in this contribution. 

The outline of the remainder of this contribution is as follows. In Section 3.7.2, we discuss radical anion dissociation in solution, in which a conical intersection has an important impact on the ground state reaction barrier, rate constant and reaction path, all of which are also influenced by nonequilibrium solvation. The excited electronic state conical intersection problem for the cis-trans isomerization of a model protonated Schiff base in solution is discussed in Section 3.7.3, focusing on the approach to, and passage through, the conical intersection, and the influence of nonequilibrium solvation thereupon. Some concluding remarks are offered in Section 3.7.4. We make no attempt to give a complete discussion for these topics, but rather focus solely on several highlights. Similarly, the references herein are certainly incomplete. We refer the interested reader to refs [1-9] for much more extensive discussions and references. 

How would these paths differ if one assumed that equilibrium solvation (ES) applied rather than nonequilibrium solvation (NES), i.e. if one ignored any solvent dynamical effects This ES condition is imposed by requiring the free energy with respect to the solvent coordinate s is a minimum at each value of r and 8, so that s is equal to its equilibrium value jeq(r, 8) ... 

To conclude on the issue of nonequilibrium solvation effects on the radical anion dissociation, while these are not very important for the rate constant itself, for the reasons just given, they are quite significant for the reaction paths as discussed above. 

One should note that the MEPs shown are not true dynamical paths, which of course can only be obtained by dynamical calculations. We have carried these out [6,9] using several different dynamical descriptions, including surface hopping trajectories [95,96]. The resulting dynamical path for the slow solvent is reasonably similar to the MEP, but this is not the case for the fast solvent, a point to which we return below. A further dynamical study [6] has compared, for the fast solvent case using surface hopping trajectories, the dynamics with the present nonequilibrium solvation description to those when equilibrium solvation is assumed. This is the most favorable case for the validity... 

Figure 2 Illustration of nonequilibrium solvation for the simple reaction model of a Eckart potential barrier representing the solute coupled linearly to a single harmonic oscillator representing the solvent. The thin curves are equipotential contours as a function of solute coordinate and solvent coordinate. The dashed line is the equilibrium solvation path for this model. The thick lines are the conventional transition slate dividing surfaces for the gas-phase reaction (vertical line that is defined in terms of the solute coordinate only) and for the solution-phase (line that makes a 28° angle with the abscissa)...

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