Energy expansion, property derivatives from

There are situations where exact equilibrium structures must be used. The most conspicuous is for the calculation of vibrational frequencies, as well as thermodynamic properties such as entropies obtained from calculated frequencies. As already discussed in Chapter 7, this is because the frequencies derive from the second derivative term, E", in a Taylor series expansion of the total energy. 

Knowing the molecular permanent multipole moments and transition moments (or closure moments derived from sum rules, such as (36)), the computation of the fust and second order interaction energies in the multipole expansion becomes very easy. One just substitutes all these multipole properties into the expressions (16), (20), (21) and (22), together with the algebraic coefficients (24) (tabulated up to terms inclusive in ref. in a somewhat different form ), and one calculates the angular functions (lb) for given orientations of the molecules. 

A natural way to introduce equations for excited states into a quantum chemical approach is to consider stimulating the molecule by a time-varying electric field to which the molecule can respond by excitation, and derive solutions from the time-dependent Schroedinger equation. Analysis then leads to equations for the excitation energies and properties of the excited state eigensolutions like transition moments. In particular, such an approach, after a Fourier transformation from time to frequency, will yield the dynamic polarizability whose spectral expansion is... 

More importantly, to obtain the Taylor series expansion one must know the equilibrium geometry of the molecule, which is one of the properties we wish to derive from our force field calculations in the first place The Taylor series expansion is also just a mathematical way of describing the potential energy surface near an equilibrium. It does not capture the physics of interatomic interactions. 

The coexistence of these two effects makes dilute near-critical solutions challenging to model properly. For modeling purposes, this characterization into solute-induced and compressibility-driven effects usually takes the form of asymptotic expressions for the mixture s Helmholtz free energy and its temperature, volume, and composition derivatives around the solvent s critical point. The usefulness of the resulting asymptotic expressions (from the residual Helmholtz free energy expansion, A"(T,p,x)y where T is temperature, p is density and jc is composition) resides in their mathematical simplicity They only require the pure solvent thermophysical properties and the critical value of the derivative dP/dXij) ... 

The translation-invariant decomposition (9.38) was first written by Post [95] and was rediscovered independently in refs. [85,86], The result (9.39) clearly constitutes an improvement with respect to the previous inequality (9.18) because the constituent mass in is decreased by a factor 3/4, and therefore the energy E. is algebraically increased. For an attractive power-law potential e(/3)r, this provides a factor (4/3). A numerical comparison is shown in table 9.1, where are listed the naive lower limit (9.18), the improved lower limit (9.39), the exact energy obtained by a hyperspherical expansion, and the variational bound derived from a Gaussian trial wave function. It is worth noticing that the new lower limit (9.39) becomes exact in the case of the harmonic oscillator. This is true for an arbitrary number A of bosons and the harmonic oscillator is the only potential for which the inequality is saturated. A beautiful proof of this property has been given by Wu and is included in ref. [ ]. 

A conceptually different approach to the calculaticHi of interfacial tensions is the use of the generalized square-gradient approach as embodied in the work of Cahn and Hilliard [216]. The Cahn-Hilliard theory provides a means for relating a particular equation of state, based on a specific statistical mechanical model, to surface and interfacial properties. The local free energy, g, in a region of nonuniform composition will depend on the local composition as well as the composition of the immediate environment. Thus, g can be expressed in terms of an expansion in the local composition and the local composition derivatives. Use of an appropriate free energy expression derived from statistical mechanics permits calculation of the surface or interfacial tension. 

Pure thermodynamics is developed, without special reference to the atomic or molecular structure of matter, on the basis of bulk quantities like internal energy, heat, and different types of work, temperature, and entropy. The understanding of the latter two is directly rooted in the laws of thermodynamics— in particular the second law. They relate the above quantities and others derived from them. New quantities are defined in terms of differential relations describing material properties like heat capacity, thermal expansion, compressibility, or different types of conductance. The final result is a consistent set of equations and inequalities. Progress beyond this point requires additional information. This information usually consists in empirical findings like the ideal gas law or its improvements, most notably the van der Waals theory, the laws of Henry, Raoult, and others. Its ultimate power, power in the sense that it explains macroscopic phenomena through microscopic theory, thermodynamics attains as part of Statistical Mechanics or more generally Many-body Theory. 

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