Free energy vibrational

The harmonic approximation can also be used to provide an estimate of the vibrational free energy, using (Refs. 1 and 6). 

The Gibbs free energy G and the chemical potentials include contributions from the internal energy, vibrational free energy, and configurational entropy. Since most relevant stmctures will have a low surface free energy, we obtain from (5.4) that... 

The treatment above assumes that the pressure and corresponding strains are entirely mechanical in origin. However, at finite temperatures there will be a kinetic pressure arising from the changes in the vibrational free energy with volume. These may be written as ... 

Figure 2. (a) Schematic illustration of the adiabatic (solid) and diabatic (dashed) vibrational free energy curves as functions of the solvent coordinate Zp for a symmetric single proton transfer reaction, (b) Potential energy curves as functions of the proton coordinate r, for three specific values of the solvent coordinate Zp indicated in (a). 

The previous work of Cukier and coworkers [7, 12] differs from the formulation described in this chapter in a number of fundamental ways. In contrast to the multistate continuum theory described in this chapter, Cukier and coworkers did not calculate mixed electronic/proton vibrational free energy surfaces as functions of two solvent coordinates. Instead, they calculated solvated proton potentials obtained by the assumption that the inertial polarization of the solvent responds instantaneously to the proton position. (This is the limit opposite to the standard adiabatic limit of the fast proton vibrational motion responding instantaneously to... 

F[E° A/A ) — °(D/D )], and AGei = A ei 0 F is Faraday s constant and F°(X/X ) is the standard electrode potential for the X/X redox couple. In this limit, the vibrational-free energy difference between the nuclear coordinates of the excited-state free energy minimum and those of the ground-state free energy minimum,... 

The vibrational free energy per molecule according to classical mechanics is calculated by the use of the density of state, h cu), equation (9). The free energy differences, thereby obtained, between empty hydrate II and ice Ic, fJ-h — — g - - u°) are given in Table 4. The free energy... 

For pure water ice, we assume that the vibrational free energy of the various H-bond isomers is nearly the same, and that quantum effects on free energy differences between isomers can be neglected. The excellence of these approximations is confirmed by the fact that H2O/D2O isotope effects have very little effect on the transition temperature of H-bond order/disorder transitions. Even the... 

As indicated, the vibrational free energy requires an average over the restricted basin set, and 0a (/I) is the statistically preferred depth in that set obtained by maximizing this modified expression. Of course, even this extension (19) breaks down at and below a glass transition temperature. 

Because the crystal is confined to a single basin, oF) = p. If the intrabasin vibrational free energy depends only on temperature, the above equation simplifies to... 

In sections 5.22 and 5.24, we have made schematic evaluations of the nature of the vibrational spectrum (see fig. 5.5). At this point, it is convenient to construct approximate model representations of the vibrational spectrum with the aim of gleaning some insight into how the vibrational free energy affects material properties such as the specific heat, the thermal expansion coefficient and processes such as structural phase transformations. One useful tool for characterizing a distribution such as the vibrational density of states is through its moments, defined by... 

High-Temperature Expansion for Vibrational Free Energy Obtain a high-temperature expansion for the free energy due to thermal vibrations given in eqn (3.110). 

The vibrational free energy of the optimized part of the ah initio region is calculated by the Partial Hessian Vibrational Analysis (PHVA) method [58]. The solvation energy (AGs) is calculated with the EFP/PCM interface developed by Bandyopadyay et al. [59] for small molecules and extended to macromolecules by Li, Pomelli, and Jensen [60,61]. 

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