Free energy surface models

The approach presented above is referred to as the empirical valence bond (EVB) method (Ref. 6). This approach exploits the simple physical picture of the VB model which allows for a convenient representation of the diagonal matrix elements by classical force fields and convenient incorporation of realistic solvent models in the solute Hamiltonian. A key point about the EVB method is its unique calibration using well-defined experimental information. That is, after evaluating the free-energy surface with the initial parameter a , we can use conveniently the fact that the free energy of the proton transfer reaction is given by... 

Figure 14-6. Two-dimensional free energy surfaces for in-line monoanionic mechanisms for the (A) un-catalytic model reaction in solution, and the catalytic (B) O p and (C) O2p pathways in the hairpin ribozyme. fi is defined as Rp-o5, R02,-P> and 2 is R02,-H2, rOnb-H2, for fl < 0.0 A and r05,-H2, rOnb-H2, f°r 1 > 0.0 A, where Opjg is for the G p proton transfer in (B), and for the O2p proton transfer in (A) and (C), respectively. The units for free energies and distances are kcal/mol and A, respectively...

In the basic model presented above, it was assumed that the hydrogen atom in the adsorbed state is neutral and weakly influences the state of the medium molecules. In this model the free energy surfaces of the solvent, determining the activation free energy of the transition between two fixed vibrational states of the proton, m and n were of parabolic shape,... 

Compared to US and its subsequent variants, the ABF method obviates the a priori knowledge of the free energy surface. As a result, exploration of is only driven by the self-diffusion properties of the system. It should be clearly understood, however, that while the ABF helps progression along the order parameter, the method s efficiency depends on the (possibly slow) relaxation of the collective degrees of freedom orthogonal to . This explains the considerable simulation time required to model the dimerization of the transmembrane domain of glycophorin A in a simplified membrane [54],... 

Such a method has seldom been used with systems containing an aqueous fluid, probably because the complexity of the solution s free energy surface and the wide range in aqueous solubilities of the elements complicate the numerics of the calculation (e.g., Harvie el al., 1987). Instead, most models employ a procedure of elimination. If the calculation described fails to predict a system at equilibrium, the mineral assemblage is changed to swap undersaturated minerals out of the basis or supersaturated minerals into it, following the steps in the previous chapter the calculation is then repeated. 

The classical Morse curve model of intramolecular dissociative electron transfer, leading to equations (3.23) to (3.27), involves the following free energy surfaces for the reactant (Grx-) and product (Gr +x ) systems, respectively ... 

Fig. 2. Wavepacket trajectories on the excited free energy surface for a two Debye modes model. The term crossing line A and B are for weakly and moderately exergonic CR, respectively.

We developed the Analytical Generalized Born plus Non-Polar (AGBNP) model, an implicit solvent model based on the Generalized Born model [37-40,44, 66] for the electrostatic component and on the decomposition of the nonpolar hydration-free energy into a cavity component based on the solute surface area and a solute-solvent van der Waals interaction free energy component modeled using an estimator based on the Born radius of each atom. 

The two parameters, and AFg, actually fully define the parabolic ET free energy surfaces Fi(X) in the MH formulation (Eigure 2). Calculation of these two parameters has become the main historical focus of the ET models addressing the thermodynamics of the ET activation barrier. The latter, according to MH theory, can be written in terms of AFg and as... 

When the number of electronic states can be limited to two (two-state model), the analytic properties of the generating function for the two CT free energy surfaces can be used to establish a linear relation between them. The 5-function in Eq. [18] can be represented as a Fourier integral that allows one to rewrite the CT free energy in the integral form... 

Clearly, the MFI description does not capture all possible complicated mechanisms of ET activation in condensed phases. The general question that arises in this connection is whether we are able to formulate an extension of the mathematical MH framework that would (1) exactly derive from the system Hamiltonian, (2) comply with the fundamental linear constraint in Eq. [24], (3) give nonparabolic free energy surfaces and more flexibility to include nonlinear electronic or solvation effects, and (4) provide an unambiguous connection between the model parameters and spectroscopic observables. In the next section, we present the bilinear coupling model (Q model), which satisfies the above requirements and provides a generalization of the MH model. 

The inequalities in Eq. [75] also define the condition for the generating function (Eq. [23]) to be analytic in the integration contour in Eq. [25]. This condition is equivalent to the linear connection between the diabatic free energy surfaces, Eq. [24]. The Q model solution thus explicitly indicates that the linear relation between the diabatic free energy surfaces is equivalent to the condition of thermodynamic stability of the collective nuclear mode driving ET. 

Figure 15 Adiabatic free energy surfaces F (Y ) in the present model (solid lines, Eqs. [105] and [106]) and in the Marcus-Hush formulation (long-dashed lines, Eqs. [41] and [119]) for self-exchange CT with AF = APf = 0, X = = 1 eV, AE12 = 0.2 eV,...

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