First moments of the liquid phase far-infrared absorption cross section and the R(ω) representation of depolarized Rayleigh scattering

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Abstract
A calculation of the first moment or average frequency ω̄ of the liquid phase far‐infrared absorption cross section is presented. The theory also applies to the calculation of the average frequency of the R(ω) representation of depolarized Rayleigh scattering data, where R(ω)=ω[1−exp(−ℏω/kT)]I(ω). The average frequency is shown to depend on the integral of the imaginary part of the angular derivative correlation function divided by time. A semiclassical relationship between the real and imaginary parts of the correlation function is derived and shown to lead to a series representation of ω̄ in terms of fourth and higher moments of the intensity spectrum. It is demonstrated that the theory reproduces the exact semiclassical result for ω̄ of both the far‐IR absorption spectrum and the R(ω) representation of the depolarized Rayleigh wing of linear free rotors. The implications of our theory concerning the comparison of peak frequencies in the far‐IR absorption spectrum and the R(ω) representation of the depolarized Rayleigh scattering spectrum are discussed. The theory is able to qualitatively explain the experimentally observed dependence of peak frequency on temperature, concentration, and isotopic substitution. The ratio of the average frequencies of the absorption cross section and the R(ω) representation is shown to depend on the size of the mean square torque compared to (kT)2.

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