Second and third binary moment

6 Molecular gases 2 Second and third binary moment 

For the rototranslational spectra, within the framework of the isotropic interaction approximation, the expressions for the zeroth and first moments, Eqs. 6.13 and 6.16, are exact provided the quantal pair distribution function (Eq. 5.36) is used [314]. A similar expression for the binary second translational moment has been reported [291], 

The superscripts I- TV in this expression designate the first through fourth derivatives of potential and dipole function with respect to R. 

The pair distribution function g(0) is similar to Eq. 5.36, except that we now have to consider the possibility of like pairs. Specifically, we write 

For dissimilar pairs, the parameter ys equals zero and we have Eq. 5.36. Like pairs of zero spin are bosons and all odd-numbered partial waves are ruled out by the requirement of even wavefunctions of the pair this calls for ys = 1. In general, for like pairs, the symmetry parameter ys will be between -1 and 1, depending on the monomer spins (fermions or bosons) and the various total spin functions of the pair. A simple example is considered below (p. 288ff.). If vibrational states are excited, the radial wavefunctions xp must be obtained from the vibrationally averaged potential, Fq(R). The functions gf(R) and gM(R) are similar to the pair distribution function, namely [294] 

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

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