Average solute-solvent energy

Unfortunately, evaluating this formula exactly would still require that we know the fully coupled solute-solvent dynamics because it calls for Fext(t) = Fext(q((t)), but since the solvent perturbs the solute vibration only weakly, a perturbative treatment suffices (just as it does quantum mechanically). To leading order, Fext(t) = F Oj, what the solvent force would be if the solute s vibrational mode were held fixed. Thus, the average rate of solute-solvent energy transfer in the steady state is... 

As long as the normalization condition given by Eq. (5) is satisfied, an arbitrary offset constant may be added to W(X) without affecting averages in Eq. (3). The absolute value of the PMF is thus unimportant. For convenience, it is possible to choose the value of the free energy W(X) relative to a reference system from which the solute-solvent interactions are absent. The free energy W(X) may thus be expressed as... 

Dispersion Interactions. Last but not least in the range of solute-solvent electrostatic interactions come the dispersion forces which depend on the polarizabilities of the molecules. Any atom or molecule—non-polar or polar—has a small fluctuating dipole moment as the electrons move around the nuclei. These instantaneous dipoles induce dipole moments in all other polarizable molecules, so that the interaction energy is proportional to the product of the average polarizabilities aM and as of the solute and solvent molecules... 

An interesting aspect of cooperativity, and one that has seldom been probed, is how this property is affected by solvation. In order to examine this question, the various H-bonded chains were next immersed in various solvents, each represented by a continuum with the dielectric constant e characteristic of that solvent. Table 15-7 reports the average H-bond energy (computed as the energy required to break the n-mer apart into n separate monomers, divided by the number of H-bonds in the chain) for dielectric constants varying from unity (vacuum) up to 78 to represent aqueous solution. 

In the currently used version of the method of energy representation [15,16,19], the solvation free energy A/x is approximately expressed in terms of distribution functions constructed from pe in the solution and pure solvent systems. In our treatments, the solution system refers to the system in which the solute molecule interacts with the solvent under the solute-solvent interaction v of interest at full coupling. In the solution, the average distribution pe of the v value is relevant in the approximate construction of Ap, and is given by... 

By virtue of Eq. (17-37), the average sum of the solute-solvent interaction energy in the solution system of interest is smaller than or equal to A/x. This means that the density-functional Fe is always non-positive for any solute-solvent distribution function. Actually, the density functional is a measure of the difference between... 

System Solvation free energy A/t Average sum of the solute—solvent interaction... 

The statistical average over the electronic degrees of freedom in Eq. [15] is equivalent, in the Drude model, to integration over the induced dipole moments pg and py. The Hamiltonian H, is quadratic in the induced dipoles, and the trace can be calculated exactly as a functional integral over the fluctuating fields pg and The resulting solute-solvent interaction energy... 

Suppose that we were to average out the effects of all of the solvent molecules, effectively integrating over the coordinates describing the solvent molecules. This would dramatically simplify the description of the solvent molecules, and thereby simplify the computation of the energy of the solute-solvent system. This is the general principle behind the implicit solvent models. The solvent is described by a single term, its dielectric constant, and we just need to treat the interaction of the solute with this field. 

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