Perturbation theory for liquids

In considering perturbation theory for liquids it is convenient first to discuss the historical development for atomic liquids. 

Using the method described in Sect. 8.3.3.1 to obtain the electric field in a microtube resonator, the device sensitivity can be calculated with the perturbation theory for different modes. This provides more systematic insight into the operation of microtube resonators. First, bulk index sensing is considered. The sensitivity is proportional to the integrated optical field inside the liquid core region over that of the entire space, and can be expressed as72 ... 

Tarazona and Navascues have proposed a perturbation theory based upon the division of the pair potential given in Eq. (3.5.1). In addition, they make a further division of the reference potential into attractive and repulsive contributions in the manner of the WCA theory. The resulting perturbation theory for the interfacial properties of the reference system is constructed through adaptation of a method developed by Toxvaerd in his extension of the BH perturbation theory to the vapor-liquid interface. The Tarazona-Navascues theory generates results for the Helmholtz free energy and surface tension in addition to the density profile. Chacon et al. have shown how the perturbation theories based upon Eq. (3.5.1) may be developed by a series of approximations within the context of a general density-functional treatment. 

The application of this approach to the hard-sphere system was presented by Ree and Hoover in a footnote to their paper on the hard-sphere phase diagram. They made a calculation where they used Eq. (2.27) for the solid phase and an accurate equation of state for the fluid phase to obtain results that are in very close agreement with their results from MC simulations. The LJD theory in combination with perturbation theory for the liquid state free energy has been applied to the calculation of solid-fluid equilibrium for the Lennard-Jones 12-6 potential by Henderson and Barker [138] and by Mansoori and Canfield [139]. Ross has applied a similar approch to the exp-6 potential. A similar approach was used for square well potentials by Young [140]. More recent applications have been made to nonspherical molecules [100,141] and mixtures [101,108,109,142]. 

Although most of the studies of this model have focused on the fluid phase in connection with the theory of electrolyte solutions, its solid-fluid phase behavior has been the subject of two recent computer simulation studies in addition to theoretical studies. Smit et al. [272] and Vega et al. [142] have made MC simulation studies to determine the solid-fluid and solid-solid equilibria in this model. Two solid phases are encountered. At low temperature the substitutionally ordered CsCl structure is stable due to the influence of the coulombic interactions under these conditions. At high temperatures where packing of equal-sized hard spheres determines the stability a substitutionally disordered fee structure is stable. There is a triple point where the fluid and two solid phases coexist in addition to a vapor-liquid-solid triple point. This behavior can be qualitatively described by using the cell theory for the solid phase and perturbation theory for the fluid phase [142]. Predictions from density functional theory [273] are less accurate for this system. 

Gross, J., Sadowski, G., Perturbed-chain SAFT An equation of state based on a perturbation theory for chain molecules, oA. Eng. Chm. Res. 40 (2001) 1244-1260. Chapman, W.G., Gubbins, K.E., Jackson, G., Radosz, M., New reference equation of state for associating liquids, Ind. Eng. Chem. Res. 29 (1990) 1709-1721. Diamantonis, N.I., Boulougouris, G.C., Mansoor, E., Tsangaris, D.M., Economou, LG. Evaluation of cubic, SAFT, cmd PC-SAFT equations of state for the vapor-liquid equilibrium modeling of CO2 mixtures with other gases, Ind. Eng. Chem. Res. 52 (2013) 3933-3942. 

In perturbation theory for simple liquids, an irreducible cluster integral (see section 1.8)... 

Relation (3.183) is useful whenever we know the free energy of the unperturbed system and when the perturbation energy is small compared with kT. It is clear that if we take more terms in the expansion (3.179), we end up with integrals involving higher-order molecular distribution functions. Therefore, such an expansion is useful only for the cases discussed in this section. For a recent review on the application of perturbation theories to liquids see Barker and Henderson (1972). 

T. Head-Gordon and F. H. Stillinger, J. Chem. Phys., 98, 3313 (1993). An Orientational Perturbation Theory for Pure Liquid Water. 

Calculations of atomic and molecular hyperpolarizabilities usually proceed via time-dependent perturbation theory for the perturbed atomic states. Even for molecules of modest size, the calculation of the complete set of unperturbed wavefunctions, and exact calculation of the hyperpolarizabilities, is prohibitively difficult. Liquid crystals typically consist of organic molecules with aromatic cores, and there is considerable experimental [10] and theoretical [11, 12] evidence to indicate that the dominant contribution to the polarizabilities originates from the delocalized r-electrons in conjugated regions of these molecules. Even considering only r-electrons the calculations rapid-... 

The effects of pressure and temperature on the physico-chemical properties of PILs were also examined.Different theoretical predictive models were developed on the subject. Noteworthy were theoretical investigations on the cation-anion interactions in PILs compared to ammonium ionic liquids, and an all-atomistic force field for a new class of halogen-free chelated orthoborate PILs. " A study was also reported on an ion contribution equation of state based on electrolyte perturbation theory for the calculation of ILs densities. ... 

Another important application of perturbation theory is to molecules with anisotropic interactions. Examples are dipolar hard spheres, in which the anisotropy is due to the polarity of tlie molecule, and liquid crystals in which the anisotropy is due also to the shape of the molecules. The use of an anisotropic reference system is more natural in accounting for molecular shape, but presents difficulties. Hence, we will consider only... 

Barker J and Henderson D 1967 Perturbation theory and equation of state for a fluids II. A successful theory of liquids J. Chem. Phys. 47 4714... 

One possibility for this was demonstrated in Chapter 3. If impact theory is still valid in a moderately dense fluid where non-model stochastic perturbation theory has been already found applicable, then evidently the continuation of the theory to liquid densities is justified. This simplest opportunity of unified description of nitrogen isotropic Q-branch from rarefied gas to liquid is validated due to the small enough frequency scale of rotation-vibration interaction. The frequency scales corresponding to IR and anisotropic Raman spectra are much larger. So the common applicability region for perturbation and impact theories hardly exists. The analysis of numerous experimental data proves that in simple (non-associated) systems there are three different scenarios of linear rotator spectral transformation. The IR spectrum in rarefied gas is a P-R doublet with either resolved or unresolved rotational structure. In the process of condensation the following may happen. 

As we have already pointed out, the theoretical basis of free energy calculations were laid a long time ago [1,4,5], but, quite understandably, had to wait for sufficient computational capabilities to be applied to molecular systems of interest to the chemist, the physicist, and the biologist. In the meantime, these calculations were the domain of analytical theories. The most useful in practice were perturbation theories of dense liquids. In the Barker-Henderson theory [13], the reference state was chosen to be a hard-sphere fluid. The subsequent Weeks-Chandler-Andersen theory [14] differed from the Barker-Henderson approach by dividing the intermolecular potential such that its unperturbed and perturbed parts were associated with repulsive and attractive forces, respectively. This division yields slower variation of the perturbation term with intermolecular separation and, consequently, faster convergence of the perturbation series than the division employed by Barker and Henderson. 

M = (Mx,My,M ) is the dipole moment of the system. Moreover, the indices i, j designate the Cartesian components x, y, z of these vectors, ()q realizes an averaging over all possible realizations of the optical field E, and () realizes an averaging over the states of the nonperturbed liquid sample. Two three-time correlation functions are present in Eq. (4) the correlation function of E(t) and the correlation function of the variables/(q, t), M(t). Such objects are typical for statistical mechanisms of systems out of equilibrium, and they are well known in time-resolved optical spectroscopy [4]. The above expression for A5 (q, t) is an exact second-order perturbation theory result. 

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