Ternary dipoles

The induced dipole of a gas sample is a function of the internuclear distances, r, the molecular orientations, Q and the positions / , of the N molecules in the gas, 

The dipole moment induced in a molecule, or in a group of molecules, is a finite range function of the intermolecular separations, R, which falls off faster than R-3 for R — oo. Van Kranendonk has argued that, therefore, it is possible to expand the above equation in a series of cluster functions [400, 402]. If ft( 1 n) designates the dipole moment induced in the cluster of molecules 1 n when these are present alone in the given volume V, cluster functions 1/(1 n) can be defined according to 

The U functions have the property of being zero unless all of the molecules 1 n are close together, within the range of the induced dipole moments. We may solve these equations for the U functions in terms of the p functions. When the solutions U are substituted in the right-hand sides of the above equations, an identity results which represents the cluster development of the /i(l N). Specifically, the /i(12) represent the dipoles induced in pairs and /i(123) the dipole induced in three-body complexes. 

We have seen above that the pairwise-additive parts of the ternary induced dipole component are well known for a number of systems. The irreducible, ternary components, on the other hand, are poorly known, for any system. Even less is known about the N-body irreducible components of the induced dipole for N 3. Attempts exist to model three- and four-body dipole components on the basis of classical relationships, but the evidence indicates that overlap effects are important the known many-body dipole models do in general not account for quantum effects. 

These facts suggest that the excess absorption is due to irreducible, ternary dipole components. We mention that classical estimates of the long-range, irreducible component in hydrogen (where quadrupolar induction prevails) have suggested a weak reduction of the absorption [402]. The observed excess absorption is, therefore, believed to arise from a different, non-classical mechanism. Overlap induction seems to be the most 

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Very little is known about the irreducible ternary dipole components. An early estimate based on classical electrodynamics, hard-sphere interaction and other simplifying assumptions suggests very small, negative contributions to the zeroth spectral moment [402], namely —0.13 x 10-10 cm-1 amagat-3. 

The comparison of Eqs. (1,2,3) shows that the ternary dipole functions B arising from the dipole moment induced in molecules 1,2 are related to the pair dipole functions B, according to... 

The B functions for the pair dipole /z(12) couple to only a small subset of the ternary dipole components B [51, 52]. 

Non-additive ternary dipole calculations of non-overlapping molecular clusters of the type A-A-A have been reported, where A stands for H, H2, He, Ne, or Ar. The induced dipoles of clusters of the type A-A-B and A-B-C, with A, B, C, designating any one of the species H, H2, and He, were computed [58, 59]. 

Ternary spectral moments of collision-induced absorption in hydrogen gas are analyzed in the H2 fundamental band in terms of pairwise additive and irreducible contributions to the interaction-induced dipole moment, Eqs. (1 - 7) [51]. Numerical results show that irreducible dipole components, especially of the exchange quadrupole-induced ternary dipole component, are significant for agreement with spectroscopic measurements, such as ternary spectral moments (Fig. 1) [53], an observed diffuse triple transition 3<3i centered at 12,466 cm-1 [52, 54, 55], and the intercollisional dip in compressed hydrogen gas, pp. 188 -190. 

Revised material in Section 5 includes an extensive tabulation of binary and ternary azeotropes comprising approximately 850 entries. Over 975 compounds have values listed for viscosity, dielectric constant, dipole moment, and surface tension. Whenever possible, data for viscosity and dielectric constant are provided at two temperatures to permit interpolation for intermediate temperatures and also to permit limited extrapolation of the data. The dipole moments are often listed for different physical states. Values for surface tension can be calculated over a range of temperatures from two constants that can be fitted into a linear equation. Also extensively revised and expanded are the properties of combustible mixtures in air. A table of triple points has been added. 

The thermodynamic stability of a ternary GIC is also questionable. Obviously, between a bare lithium ion and one solvated by molecular dipoles, the intercalation of the former between two giant graphene anions is far more favored thermodynamically than that of the latter. The fully lithiated GIC LiCe, for example, does not solvate in nonaqueous electrolyte solvents, and the tendency of lithium to prefer binary (i.e., without solvent cointercalation) instead of ternary GICs has also been noticed in the solution syntheses. 

Binary and ternary spectra. We will be concerned mainly with absorption of electromagnetic radiation by binary complexes of inert atoms and/or simple molecules. For such systems, high-quality measurements of collision-induced spectra exist, which will be reviewed in Chapter 3. Furthermore, a rigorous, theoretical description of binary systems and spectra is possible which lends itself readily to numerical calculations, Chapters 5 and 6. Measurements of binary spectra may be directly compared with the fundamental theory. Interesting experimental and theoretical studies of various aspects of ternary spectra are also possible. These are aimed, for example, at a distinction of the fairly well understood pairwise-additive dipole components and the less well understood irreducible three-body induced components. Induced spectra of bigger complexes, and of reactive systems, are also of interest and will be considered to some limited extent below. 

Content. After a brief overview of molecular collisions and interactions, dipole radiation, and instrumentation (Chapter 2), we consider examples of measured collision-induced spectra, from the simplest systems (rare gas mixtures at low density) to the more complex molecular systems. Chapter 3 reviews the measurements. It is divided into three parts translational, rototranslational and rotovibrational induced spectra. Each of these considers the binary and ternary spectra, and van der Waals molecules we also take a brief look at the spectra of dense systems (liquids and solids). Once the experimental evidence is collected and understood in terms of simple models, a more theoretical approach is chosen for the discussion of induced dipole moments (Chapter 4) and the spectra (Chapters 5 and 6). Chapters 3 through 6 are the backbone of the book. Related topics, such as redistribution of radiation, electronic collision-induced absorption and emission, etc., and applications are considered in Chapter 7. 

Besides this intercollisional interference process, there are other three-body processes which at elevated densities affect the observable spectra over a much wider range of frequencies, virtually at all frequencies at which absorption may be observed. With increasing density, one will be able to discern binary, ternary, and perhaps higher-order spectral contributions (even if cvxi2 1). These are caused by the dipoles induced in systems consisting of N interacting atoms or molecules, with N > 2. 

Table 3.7 also lists ternary spectral moments for a few systems other than H2-H2-H2. For the H2-He-He system, the pairwise-additive dipole moments are also known from first principles. The measured spectral moments are substantially greater than the ones calculated with the assumption of pairwise additivity - just as this was seen in pure hydrogen. For the other systems listed in the Table, no ab initio dipole surfaces are known and a comparison with theory must therefore be based on the approximate, classical multipole model. 

Besides the collision-induced dipoles, we will occasionally refer to field-induced dipoles, or to rotation-induced dipoles, that is dipoles induced by an external electric field, or by centrifugal forces distorting certain symmetries of rotating molecules. Moreover, we will be interested in the dipoles induced in binary, ternary, etc., systems as we proceed. 

In order to estimate the strength of the ternary, irreducible dipole components, Guillot et al. used the one effective electron model [171, 173]. For H2-He-He at liquid state densities, the (admittedly crude) model suggests significant enhancement of the absorption due to overlap-induced, irreducible dipole components in molecular dynamics studies, in qualitative agreement with the above conclusion. 

Near-range components. The dipole moments of three like atoms at near range were estimated using variational methods and a one-electron model. The results are believed to represent the qualitative features of the overlap-induced triatomic induced dipole, which is the dominant dipole at the spectroscopically important near range [1, 173], The triatomic overlap-induced dipoles were estimated to be just slightly weaker than the binary ones [1], The observation of very weak ternary absorption would then be a consequence of strong cancellations of long- and short-term induced dipole components [172]. 

Early numerical estimates of ternary moments [402] were based on the empirical exp-4 induced dipole model typical of collision-induced absorption in the fundamental band, which we will consider in Chapter 6, and hard-sphere interaction potentials. While the main conclusions are at least qualitatively supported by more detailed calculations, significant quantitative differences are observed that are related to three improvements that have been possible in recent work [296] improved interaction potentials the quantum corrections of the distribution functions and new, accurate induced dipole functions. The force effect is by no means always positive, nor is it always stronger than the cancellation effect. 

Helium-argon mixtures. For the He-Ar pair, an accurate ab initio induced dipole surface exists, Table 4.3 which, with the help of line shape calculations, was shown to reproduce the binary collision-induced absorption spectra within the accuracy of the measurement [278]. For the ternary moments, the SPFD2 He-Ar [12] and the HFD-C Ar-Ar [11] interaction potentials were input, along with this ab initio dipole surface. 

Density expansion. The method of cluster expansions has been used to obtain the time-dependent correlation functions for a mixture of atomic gases. The particle dynamics was treated quantum mechanically. Expressions up to third order in density were given explicitly [331]. We have discussed similar work in the previous Section and simply state that one may talk about binary, ternary, etc., dipole autocorrelation functions. 

Ternary moments have been computed for several systems of practical interest [314, 422]. Recent studies are based on accurate ab initio pair dipole surfaces obtained with highly correlated wavefunctions. Because not much is presently known about the irreducible ternary components, it is important to determine to what extent the measured three-body spectral components arise from pairwise-additive contributions [296, 299]. 

Hydrogen. Accurate ab initio dipole surfaces for both the rototrans-lational collision-induced absorption spectrum in the far infrared [282], and the rotovibrational collision-induced absorption spectrum in the near infrared [281] have been obtained that may be used to compute the pairwise-additive ternary contributions for both bands. However, measurements of the density-dependence exist only for the fundamental band, Table 6.6. We note that according to the nomenclature adopted, molecule 1 undergoes the vibrational transition. 

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