Equilibrium solvation path approximation

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. 

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. 

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]. 

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]. 

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 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. 

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