Equilibrium solvation path

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... 

In the equilibrium solvation path (ESP) approximation [74, 76], ve first find a potential of mean force surface for the primary subsystem in the presence of the secondary subsystem, and then we finish the calculation using this free energy surface. Notice a critical difference from the SES in that now we find the MEP on U rather than V, and we now find solute vibrational frequencies using U rather than V. 

If the reaction path and dividing surface are optimized in the gas phase, but the rate constant is calculated with the equilibrium solvation Hamiltonian, the resulting rate constant is called separable equilibrium solvation (SES) [57]. However, if the reaction path and dividing surface are optimized with the equilibrium solvation potential, the result is labeled equilibrium solvation path (ESP) [57,78]. 

Figure 2 illustrates the model of a single effective. solvent coordinate for the simple case of a single harmonic oscillator coupled to an Eckart potential. The equilibrium solvation path for this model is obtained by finding the minimum in the potential with respect to the solvent coordinate at each location... 

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)...

Because the transition state geometry optimized in solution and the solution-path reacton path may be very different from the gas-phase saddle point and the gas-phase reaction path, it is better to follow the reaction path given by the steepest-descents-path computed from the potential of mean force. This approach is called the equilibrium solvation path (ESP) approximation. In the ESP method, one also substitutes W for V in computing the partition functions. In the ESP approximation, the solvent coordinates are not involved in the definition of the generalized-transition-state dividing surface, and hence, they are not involved in the definition of the reaction coordinate, which is normal to that surface. One says physically that the solvent does not participate in the reaction coordinate. That is the hallmark of equilibrium solvation. 

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. 

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. 

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) ... 

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... 

The simplest way to include solvation effects is to calculate the reaction path and tunneling paths of the solute in the gas phase and then add the free energy of solvation at every point along the reaction path and tunneling paths. This is equivalent to treating the Hamiltonian as separable in solute coordinates and solvent coordinates, and we call it separable equilibrium solvation (SES) [74]. Adding tunneling in this method requires a new approximation, namely the canonical mean shape (CMS) approximation [75]. 

So there is a high cost in energy, identified as the reorganization energy, Xq, to transfer the charge if the reactants are in equilibrium with the solvent and the products are fiilly out of equihbrium. Figure 11.3. Marcus has shown that die reaction can proceed by means of a lower barrier. This path requires a fluctuation that takes the reactants away from equilibrium and part of the way toward the equilibrium solvation configuration of file products. One stiU forms products not in equihbrium, but the extent of solvation disequilibrium in the nascent products is reduced. 

Shifts in dynamical bottlenecks due to equilibrium solvation can be accounted for using variational TST. The simplest approximate method to do this is to compute the solvation free energy along the gas-phase reaction coordinate to generate an equilibrium solvation free energy of activation that is a function of the location s of the dividing surface along the reaction path ... 

The oxidation or reduction of a substrate suffering from sluggish electron transfer kinetics at the electrode surface is mediated by a redox system that can exchange electrons rapidly with the electrode and the substrate. The situation is clear when the half-wave potential of the mediator is equal to or more positive than that of the substrate (for oxidations, and vice versa for reductions). The mediated reaction path is favored over direct electrochemistry of the substrate at the electrode because, by the diffusion/reaction layer of the redox mediator, the electron transfer step takes place in a three-dimensional reaction zone rather than at the surface Mediation can also occur when the half-wave potential of the mediator is on the thermodynamically less favorable side, in cases where the redox equilibrium between mediator and substrate is disturbed by an irreversible follow-up reaction of the latter. The requirement of sufficiently fast electron transfer reactions of the mediator is usually fulfilled by such revemible redox couples PjQ in which bond and solvate... 

The RISM integral equations in the KH approximation lead to closed analytical expressions for the free energy and its derivatives [29-31]. Likewise, the KHM approximation (7) possesses an exact differential of the free energy. Note that the solvation chemical potential for the MSA or PY closures is not available in a closed form and depends on a path of the thermodynamic integration. With the analytical expressions for the chemical potential and the pressure, the phase coexistence envelope of molecular fluid can be localized directly by solving the mechanical and chemical equilibrium conditions. 

In the equilibrium-secondary-zone approximation [82, 85] we refine the effective potential along each reaction path by adding the charge in secondary-zone free energy. Thus, in this treatment, we include additional aspects of the secondary subsystem. This need not be more accurate because in many reactions the solvation is not able to adjust on the time scale of primary subsystem barrier crossing [86]. 

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