First Derivatives of the Free Energy

For a given and Ne, the Gibbs free energy is given by (6.35), which can be simplified by using the Stirling approximation 

This function has a single minimum for as shown previously. 

The system described by the variables T, P,, and must be frozen in with respect to the conversion L H, otherwise Nl is not an independent variable. 

L and H are both water molecules. However, by their very definition, they differ markedly in local environment hence, it is unlikely that they obey the condition for symmetric ideal solutions. This point will be discussed further in Section 6.8, when we analyze a more general mixture-model approach to water. [See also the discussion following (6.52).] 

This is the case when we tend to a pure lattice L. The chemical potential [jiL tends to the enthalpy (see below) of pure L, whereas the chemical potential of H tends to minus infinity. Such a system, of course, cannot be at equilibrium it may be attained in the frozen-in system. (Note that in this case, H forms a dilute ideal solution in the system.) 

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Phase transitions at which the entropy and enthalpy are discontinuous are called first-order transitions because it is the first derivatives of the free energy that are disconthuious. (The molar volume V= (d(i/d p) j is also discontinuous.) Phase transitions at which these derivatives are continuous but second derivatives of G... 

If classical Coulombic interactions are assumed among point charges for electrostatic interactions between solute and solvent, and the term for the Cl coefficients (C) is omitted, the solvated Eock operator is reduced to Eq. (6). The significance of this definition of the Eock operator from a variational principle is that it enables us to express the analytical first derivative of the free energy with respect to the nuclear coordinate of the solute molecule R ,... 

When the free energies F of the two crystal structures are identical, the system is at a critical point. The identity of F does not imply identical fimctions (otherwise the two phases would be indistinguishable). Therefore, at the critical point first derivatives of F might differ and therefore enthalpy, volume, and entropy of the two phases would be different. These transformations are first-order phase transitions, according to Ehrenfest [105]. A discontinuous enthalpy imphes heat exchange at the transition temperature, which can easily be measured with DSC experiments. A discontinuous volume is evident under the microscope or, more precisely, with diffraction experiments on single crystals or powders. Some phase transitions are however characterized by continuous first derivatives of the free energy, whereas the second derivatives (specific heat, compressibility, or thermal expansivity, etc.) are discontinuous. These transformations are second-order transitions and are clearly softer. 

All the alternative variants of the MPn may be implemented using a relaxed density matrix or a unrelaxed density matrix, in analogy with the Cl solvation methods. In the first case the correlated electronic density is computed as a first derivatives of the free energy, while in the second case only the MPn perturbative wavefunction amplitudes are necessary. 

Although the correlative methods based on the coupled-cluster (CC) ansatz are among the most accurate approaches for molecules in vacuum, their extension to introduce the interactions between a molecule and a surrounding solvent have not yet reached a satisfactory stage. The main complexity in coupling CC to solvation methods comes from the evaluation of the electronic density, or of the related observables, needed for the calculation of the reaction field. Within the CC scheme the electronic density can only be evaluated by a relaxed approach, which implies the evaluation of the first derivative of the free energy functional. As discussed previously for the cases of the Cl and MPn approaches, this leads to a more involved formalism. 

Given that (see Fig. 9.8) at the glass transition temperature, the specific volume Vs and entropy S are continuous, whereas the thermal expansivity a and heat capacity Cp are discontinuous, at first glance it is not unreasonable to characterize the transformation occurring at Tg as a second-order phase transformation. After all, recall that, by definition, second-order phase transitions require that the properties that depend on the first derivative of the free energy G such as... 

The first derivative of the free energy with respect to the nuclear coordinate of the solute molecule Ra is written as... 

The exact eigenfunctions of the effective PCM Hamiltonian (1.12) obey to a generalized Hellmann-Feynman, theorem according to which the first derivative of the free-energy functional G (1.10) with respect to a perturbation parameter k may be compute as expectation value with the unperturbed wavefunction ... 

The Helmann-Feynman theorem (2.1) implies that the expectation value of the first-order observable of the molecular solute can be expressed as first derivative of the free energy functional G with respect to a suitable perturbation. If we consider as external perturbation the operator x6 corresponding to the observable of interest d times a scalar factor A, ... 

Eq. (A. 14) shows that the sum of two-times the kinetic energy with the total potential energy, including the solute-solvent interaction, is equal to the first derivative of the free-energy functional G with respect to a uniform scaling of the Cartesian co-ordinates of the nuclei and of the position of the polarization charges. Equation. (A.13) reduces to VT case for isolated molecules if aU the solvation contribution are neglected. 

Phase transitions can be first-order or second-order (or continuous), and critical energy fluctuations quite often have a significant impact on thermal parameters. First-order transitions are characterized by discontinuous jumps in the first derivatives of the free energy, resulting in finite density p and enthalpy H differences between two distinct coexisting phases at the transition temperature Tp.. For a second-order transition there are no discontinuities in the density or the enthalpy but the specific heat capacity Cp will exhibit either a discontinuous-jump (for mean-field regime) or a critical anomaly... 

The analytical first derivative of the free energy PCM-EOM functional may be eas-... 

In the case (ii), the sign of the excess free energy is determined by the sign of the first derivative of the free energy of a single... 

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