Collective-solvent-coordinate model

Solvation Thermodynamics and the Treatment of Equilibrium and Nonequilibrium Solvation Effects by Models Based on Collective Solvent Coordinates... 

Nonequilibrium solvent effects can indeed by significant at the kcal level-maybe even at a greater level, but so far there is no evidence for that when the reaction coordinate involves protonic or heavier motions. Our goal in this section has been to emphasize just how powerful and general the equilibrium model is. In addition, in both the previous section and the present section, we have emphasized the use of models based on collective solvent coordinates for calculating both equilibrium and nonequilibrium solvation properties. 

In the absence of discrete solvent molecules or a collective solvent coordinate, continuum solvation models do not allow the solvent to enter into the reaction coordinate, and in many cases that misses the primary role of the solvent. The solvent may enter the reaction coordinate only quantitatively, for example by having a slightly different strength of hydrogen bonding to the solute at the transition state than at the reactant, or it may enter qualitatively, for example by entering or leaving the first solvation shell, by... 

Assuming a simple Marcus-type model for the interaction with the solvent, one can derive an analytical expression for the potential (free) energy surface of the bond-breaking electron-transfer reaction as a function of the collective solvent coordinate q and the distance r between the fragments R and X [71,75]. The activation energy of the reaction can also be calculated explicitly ... 

Cramer CJ, Truhlar DG. Solvation thermodynamics and the treatment of equilibrium and nonequilibrium solvation effects by models based on collective solvent coordinates. In Reddy MR, Erion MD, eds. Free Energy Calculations in Rational Drug Design. New York Kluwer/Plenum, 2001 63-95. 

Reddy and M. D. Erion, Eds., Kluwer Academic/Plenum, New York, 2001, pp. 63-95. Solvation Thermodynamics and the Treatment of Equilibrium and Nonequilibrium Solvation Effects by Models Based on Collective Solvent Coordinates. 

The Golden Rule formula (9.5) for the mean rate constant assumes the Unear response regime of solvent polarization and is completely equivalent in this sense to the result predicted by the spin-boson model, where a two-state electronic system is coupled to a thermal bath of harmonic oscillators with the spectral density of relaxation J(o)) [38,71]. One should keep in mind that the actual coordinates of the solvent are not necessarily harmonic, but if the collective solvent polarization foUows the Unear response, the system can be effectively represented by a set of harmonic oscillators with the spectral density derived from the linear response function [39,182]. Another important point we would like to mention is that the Golden Rule expression is in fact equivalent [183] to the so-called noninteracting blip approximation [71] often used in the context of the spin-boson model. The perturbation theory can be readily applied to... 

The model offers insights on several key points regarding the role of the solvent. But in the final analysis it cannot substitute for realistic dynamical simulations. In particular, the model overlooks any role played the internal modes of the solvent molecules or the internal modes of the solute. What the model provides is insight into the motion along the solvent-modified reaction coordinate. Specifically, the diffusion forward and backward across the barrier, that in the Kramers modef occurs along the reaction coordinate of the isolated solute, is shown in the model to be equivalent to a single crossing of the transition state provided that we use a collective reaction coordinate, one that involves both solute and solvent motion. 

Clearly, a general theory able to naturally include other solvent modes in order to simulate a dissipative solute dynamics is still lacking. Our aim is not so ambitious, and we believe that an effective working theory, based on a self-consistent set of hypotheses of microscopic nature is still far off. Nevertheless, a mesoscopic approach in which one is not limited to the one-body model, can be very fruitful in providing a fairly accurate description of the experimental data, provided that a clever choice of the reduced set of coordinates is made, and careful analytical and computational treatments of the improved model are attained. In this paper, it is our purpose to consider a description of rotational relaxation in the formal context of a many-body Fokker-Planck-Kramers equation (MFPKE). We shall devote Section I to the analysis of the formal properties of multivariate FPK operators, with particular emphasis on systematic procedures to eliminate the non-essential parts of the collective modes in order to obtain manageable models. Detailed computation of correlation functions is reserved for Section II. A preliminary account of our approach has recently been presented in two Letters which address the specific questions of (1) the Hubbard-Einstein relation in a mesoscopic context [39] and (2) bifurcations in the rotational relaxation of viscous liquids [40]. 

We start with the simplest model of the interface, which consists of a smooth charged hard wall near a ionic solution that is represented by a collection of charged hard spheres, all embedded in a continuum of dielectric constant c. This system is fairly well understood when the density and coupling parameters are low. Then we replace the continuum solvent by a molecular model of the solvent. The simplest of these is the hard sphere with a point dipole[32], which can be treated analytically in some simple cases. More elaborate models of the solvent introduce complications in the numerical discussions. A recently proposed model of ionic solutions uses a solvent model with tetrahedrally coordinated sticky sites. This model is still analytically solvable. More realistic models of the solvent, typically water, can be studied by computer simulations, which however is very difficult for charged interfaces. The full quantum mechanical treatment of the metal surface does not seem feasible at present. The jellium model is a simple alternative for the discussion of the thermodynamic and also kinetic properties of the smooth interface [33, 34, 35, 36, 37, 38, 39, 40]. 

The coordination number collective variable is a good choice for modeling solvent thermodynamics of ions in solution. The coordination number of species A with respect to species B is defined using a Fermi-Dirac function as... 

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