Polarization free energy

The classical Born expression for the polarization free energy of a spherical ion of net charge q can be written as82 ... 

We obtain an r.m.s. deviation of 0.84 kcal/mol with an optimal a of 0.181. One can also note the similarity between the a value of this model and that of the two-parameter model with a free a and /3. This suggests that the model is robust in the sense that the actual polar and non-polar free energy contributions are more or less invariant, as long as deviations from linear response are taken into account in a proper way. The FEP-derived model could be considered preferable to the two-parameter model since it contains only one free parameter, viz. oc. The results of adding a constant yto the new model was also investigated. Remarkably, the optimal value for such a y was found to be -0.02 kcal/mol, i.e. virtually zero. 

GB-like approximations [41, 71, 119, 161, 187, 189, 230-233] may be derived from eq (1) by using the concept of dielectric energy density, as in the work of Bucher and Porter [130], Ehrenson [131], and Schaefer and Froemmel [234], As the GB methodology has been extensively reviewed in the recent past [81, 83, 213], we confine our presentation to a very brief discussion of the key aspects of the theory. The polarization free energy in the GB model is defined as... 

We also adopt the above combination rule (Eq. [6]) for the general case of exp-6 mixtures that include polar species. Moreover, in this case, we calculate the polar free energy contribution Afj using the effective hard sphere diameter creff of the variational theory. 

Fig. 1. The Marcus parabolic free energy surfaces corresponding to the reactant electronic state of the system (DA) and to the product electronic state of the system (D A ) cross (become resonant) at the transition state. The curves which cross are computed with zero electronic tunneling interaction and are known as the diabatic curves, and include the Born-Oppenheimer potential energy of the molecular system plus the environmental polarization free energy as a function of the reaction coordinate. Due to the finite electronic coupling between the reactant and charge separated states, a fraction k l of the molecular systems passing through the transition state region will cross over onto the product surface this electronically controlled fraction k l thus enters directly as a factor into the electron transfer rate constant...

The charge density p of the solute may be expressed either as some continuous function of r or as discrete point charges, depending on the theoretical model used to represent the solute. The polarization energy, Gp, discussed above, is simply the difference in the work of charging the system in the gas phase and in solution. Thus, in order to compute the polarization free energy, all that is needed is the electrostatic potential in solution and in the gas phase (the latter may be regarded as a dielectric medium characterized by a dielectric constant of 1). 

The simplicity of the BKO approach to computing polarization free energies led to its widespread use for the qualitative analysis of solvation effects on various properties for many years (including in the absence of any explicit theoretical calculations). For quantitative purposes, however, it suffers from a number of undesirable features. One such feature is the slow nature of the convergence of Eq. (11.17) with respect to Z. Table 11.2 lists AGep... 

In order to solve die Poisson equation for an arbitrary cavity, recourse to numerical methods is required. An altemative approach that has seen considerable development involves computing die polarization free energy using an approximation to the Poisson equation that can be solved analytically, and diis is the Generalized Bom (GB) approach. As its name implies, the GB method extends the Born Eq. (11.12) to polyatomic molecules. The fundamental equation of the GB method expresses the polarization energy as... 

So, the steps in a GB calculation to determine the polarization free energy given a particular molecular geometry are essentially ... 

Of course, the response of a conductor to a solute charge distribution is complete , while that of a dielectric medium is not. So, in COSMO models, the more simply evaluated conductor-polarization free energy is scaled by a factor of 2(s — l)/(2e -f-1) after its computation (i.e., by the Onsager factor in the case of the SM5C model, however, the scaling factor is (e — l)/e - see Section 11.3.3). 

The polarization free energy of inserting a molecule in a medium of dielectric constant c is expressed in the generalized Bom approximation for multi-centered systems as ... 

To include the effect of solvent polarization in molecular mechanics, Still and co-workers turned to the generalized Born modelT i i", 194-203 this model, the electric polarization free energy is written in atomic units as... 

Figure 4 The ENP component of the potential of mean force is the sum of a positive distortion energy and a negative (but rapidly less steeply changing) polarization free energy. The solvated minimum occurs close to, but not precisely at, the minimum of AGgNp because G ds has a (typically very small) dependence on geometry.

Ion channel studies motivated Allen et al. [47] who have developed an elegant variational formalism to compute polarization charges induced on dielectric interfaces. They solved the variational problem with a steepest descent method and applied their formulation in molecular dynamics (MD) simulations of water permeation through nanopores in a polarizable membrane [48-50], Note that the functional chosen by Allen et al. [47] is not the only formalism that can be used. Polarization free energy functionals [51-53] are more appropriate for dynamical problems, such as macromolecule conformational changes and solvation [54-57],... 

A series of continuum solvation models (SMx, x = 1-5) has been developed by Truhlar and co-worker (Cramer and Truhlar [79]), based on the Generalized Born/Surface Area (GB/SA) model (Still et al. [86]). Recall that in the GB approach the molecular shape is taken into account as the solute charge is distributed over a set of atom-centered spheres. For this GB/SA model, the polarization free energy is given by... 

In addition, an important theorem I0] states that the variation of the polarization free energy, Ap is related to. W f by the simple equation... 

The very successful Bom model for a monoatomic ion immersed in a solvent expresses the polarization free energy as... 

A enp is the change in the internal electronic kinetic and electronic and nuclear coulombic energy of the solute upon relaxation in solution, which is driven by the favorable electric polarization interaction with the solvent, while Cp is the electrostatic polarization free energy expressed in terms of the generalized Bom approximation (equation 40). 

The adsorption of proteins from aqueous solutions onto hydrophobic surfaces has a simple and straightforward physico-chemical mechanism, based upon the global hydrophobic attraction between an apolar, or largely apolar surface and a hydrophilic polymer (such as a protein), driven by the polar free energy of cohesion of the surrounding water molecules cf. Chapter 9, section 9.1. Thus, the attraction between a protein molecule and a hydrophobic surface, immersed in water is appropriately designated as a macroscopic interaction. 

---