Anharmonic free energy

In the present study, we examine the magnitude of the anharmonic contribution from the host water molecules and also the anharmonic free energy arising from the rotational degrees of freedom of guest molecule by Monte Carlo (MC) simulation with the Gaussian statistics. 

The anharmonic free energy is evaluated for empty hydrate and cubic ice (ice Ic). The calculated free energy due to the anharmonic potential energy surface is given in Table 1. The anharmonic contribution to the free energy of empty... 

The free energy differences between the real and the reference clathrate hydrates are also given in Table 1. The anharmonic free energy change from the spherical guests with harmonic approximation to the nonspherical guests is -0.25... 

Table 1. Free energy due to the anbarmonic contributions. Free energy is in kj/mol. The anharmonic free energy of the guest molecule is denoted byA (kJ per mole of guest). The reference systems for ice, empty hydrate and the hydrate encaging spherical guest molecules are corresponding harmonic oscillators. The reference system for the hydrate encaging nonspberical guest molecules is the hydrate occupied by spherical guest molecules.

E. L. Pollock, Direct Monte Carlo calculation of anharmonic free energies, J. Phys. C. 9, 1129-1134 (1976). 

The chemical potential difference is tabulated in Table III [17,18]. There are small but nonmegligible differences depending on both the method and the pair potential. Two distinctive features are noteworthy. CS-II is more stable than CS-I irrelevant as to whether the anharmonic free energy is taken into account for the TIP4P potential (all the properties are calculated for clathrate hydrates with this potential unless otherwise mentioned) [52]. This is also true for most of the other pair potential of water such as SPC/E [53]. The chemical potential difference is negative for most of the potential but it is positive for the CC potential [54] and other potentials with the same functional form, which is favorable to observe a hydration structure around a hydrophobic solute but is inappropriate to evaluate the thermodynamic stability of clathrate hydrates. 

This approximation can also be easily obtained from the expression (11.29). Based on our discussion above, we expect Acm < Aqm < -4FH. Figure 11.1 illustrates these bounds. The Feynman-Hibbs free energy provides a very good representation at moderate temperatures, at which some excited states are populated. See Kleinert s book [45] for applications to anharmonic systems. 

Fig. 1.3. Upper Schematic view (dotted line) of cross-section of many-dimensional highly anharmonic potential energy surfaces for reactants plus solution (R) and (dotted line omitted) for products plus solution (P). TS occurs at the intersection. Lower Plot of free energy G for the above R and P systems vs. the reaction coordinate U.

Anharmonic higher order terms gain importance for stronger solute-solvent couplings requiring 0 in Eq. [121]. The nonequilibrium solvent polarization can be considered as an ET reaction coordinate. The curvature of the corresponding free energy surface is... 

TABLE 19.1 Gibbs free energy (at 298 K and 1 atm reference pressure) for the addition of water molecules to sulfuric acid, from various calculations and experiments. For the quantum chemical results, the use of harmonic or anharmonic vibrational frequencies has been indicated... 

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