Binary moments

Spectral moments may be computed from expressions such as Eqs. 5.15 or 5.16. Furthermore, the theory of virial expansions of the spectral moments has shown that we may consider two- and three-body systems, without regard to the actual number of atoms contained in a sample if gas densities are not too high. Near the low-density limit, if mixtures of non-polar gases well above the liquefaction point are considered, a nearly pure binary spectrum may be expected (except near zero frequencies, where the intercollisional process generates a relatively sharp absorption dip due to many-body interactions.) In this subsection, we will sketch the computations necessary for the actual evaluation of the binary moments of low order, especially Eqs. 5.19 and 5.25, along with some higher moments. 

It represents the kinetic energy of the center of mass the resulting uniform motion is of little interest here. 

The Hamiltonian of relative motion, on the other hand, is given by 

Wavefunctions of relative motion are obtained by introducing spherical coordinates R, 9, p and the partial wave expansion, according to  

The sum of the potential, V(R), and the centrifugal barrier fiV(/+l)/2mR2 is called the effective potential, 

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The binary moments, yjjab and the various ternary ones, namely y(naab>, y(nabb>, yf,aaaK yj,bbbK may thus be computed. Poll and van Kranendonk give the first moments, yi, according to... 

Second binary moment. Expressions for the second binary moment are also known [319, 292], We simply quote here the results [296],... 

Third binary moment. Poll [319] has given a formula for the second and third binary moments. The third moment may be expressed as... 

A few results of quantum moment calculations for dissimilar rare gas pairs are shown in Tables 5.1, 5.2 and 3.1, and will be discussed below. The classical fourth binary moment was also reported [79], Table 5.1 compares quantum, semi-classical and classical calculations based on sum formulae (Eqs. 5.37, 5.38, 5.39) with moments obtained by integration of computed line shapes for the He-Ar system at 295 K. 

By expressing the time derivatives in terms of the Poisson bracket and canonically averaging, we get for the second binary moment... 

This expression is squared and canonically averaged, to get the fourth binary moment [79],... 

This is the semi-classical, second, binary moment. While the zeroth and first moments require only static quantum corrections of the Wigner-Kirkwood type, the second and all higher moments require also dynamical corrections involving yC4 and higher moments. 

Molecular gases 2 Second and third binary moment... 

The binary moment relations quoted above have been successfully used for the rototranslational bands and, to some extent, also for the rotovibrational bands [342], However, it has been noted [151] that for the rotovibrational bands, all but the zeroth spectral moment are affected by the variation of the interaction potential with the vibrational excitation, an effect not accounted for in Eqs. 6.13 through 6.18, 6.21. 

Ternary moments are generally associated with greater quantum corrections than binary moments, Tables 5.2 and 6.3. Quantum corrections are most significant near the repulsive core of the interaction potential. Apparently, for three-body interactions, core penetration is more significant spectroscopically than for two-body interactions. 

R. E. Bryant and Y.-A. Chen. Verification of arithmethic functions with binary moment diagrams. Technical Report CMU-CS-94-160, Carnegie Mellon University, May 1994. 

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. 

Binary molecular co-crystals of 2,5-bis(3-pyridyl)-l,3,4-oxadiazole and 2,5-bis-(4-pyridyl)-l,3,4-oxadiazole with benzene-1,3,5-tricarboxylic and benzene-1,2,4,5-tetracarboxylic acids were studied by X-ray and thermogravimetric analysis of mass loss <2005MI1247>. Dipole moments were used to study the flexoelectric effect in guest-host mixtures of 2,5-(4-pentylbenzene)-l,3,4-oxadiazole with commercial liquid crystal hosts <2005CM6354>. The luminescence properties of many other copolymers were also investigated (see Section 5.06.12.3). 

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