Irreducible, ternary

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... 

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]. 

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. 

Theory suggests that ternary moments vary substantially with temperature even sign changes occur with modest temperature variations. This fact offers intriguing possibilities for an experimental separation of the pairwise-additive and the irreducible three-body effects and, perhaps, for a critical search for irreducible ternary contributions. 

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. 

Ternary virial coefficients consist of two parts. One part is defined in terms of the pairwise additive functions of the interaction, the other by the irreducible ones. Since the latter are not well known, a vast field of study, nearly untouched, is being opened up by collision-induced spectroscopy. 

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. 

Figure 1 Comparison of observed ( , ) and computed ternary spectral moments of collision-induced absorption of compressed hydrogen in the H2 fundamental band. The dashed curve represents the moments computed on the basis of the pairwise additive dipole components only. The solid line also accounts for the irreducible dipole components from Ref. [53]. (This figure is an update of Fig. 3.46.)...

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. 

Table 3.4 Crystal structures of some solid ternary-oxide structures with the stoichiometry ABOj, and the irreducible representations of the optical modes.

Similar expressions can be written for the mean force potentials of clusters higher than the ternary one, though not displayed here. The quantity U j is referred to as the binary cluster (interaction) energy, and Auijm and Auijmn as the residual (or irreducible) potentials of ternary and quaternary clusters, respectively. 

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