Spectral moment higher-order

Comparison of formulae (2.51) and (2.64) allows one to understand the limits and advantages of the impact approximation in the theory of orientational relaxation. The results agree solely in second order with respect to time. Everything else is different. In the impact theory the expansion involves odd powers of time, though, strictly speaking, the latter should not appear. Furthermore the coefficient /4/Tj defined in (2.61) differs from the fourth spectral moment I4 both in value and in sign. Moreover, in the impact approximation all spectral moments higher than the second one are infinite. This is due to the non-analytical nature of Kj and Kf in the impact approximation. In reality, of course, all of them exist and the lowest two are usually utilized to find from Eq. (2.66) either the dispersion of the torque (M2) or related Rq defined in Eq. (1.82) ... 

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. 

For a computation of the zeroth and first moments, this lowest-order Wigner-Kirkwood correction is often sufficient, but the second and higher spectral moments require additional dynamical corrections [177]. 

Classical and semiclassical moment expressions. The expressions for the spectral moments can be made classical by substituting the classical distribution function, g(R) = exp (— V(R)/kT), for the quantum expressions. Wigner-Kirkwood corrections are known which account to lowest order for the static quantum corrections, Eq. 5.44 [177, 292]. For the second and higher moments, dynamic quantum corrections must also be made [177]. As was mentioned in the previous Chapter, such semiclassical corrections are useful in supplementing quantum computations of the spectral moments at large separations where the quantum effects are small the computational effort of quantum calculations, which is substantial at large separations, may thus be avoided. 

Higher-order classical moments have also been reported. We mention the classical expressions for the translational spectral moments M , with n = 0, 2, 4, and 6, for pairs of linear molecules given in an appendix of [204]. Spectral moments of spherical top molecules have been similarly considered [163, 205], We note that for n > 1, spectral moments show dynamic as well as static quantum correction, which become more important as the order n of the spectral moments is increased. The discussions on pp. 219, and Table 5.1, suggest that, even for the near-classical systems, quantum corrections may be substantial and can rarely be ignored. 

Spectral moments can also be computed from classical expressions with Wigner-Kirkwood quantum corrections [177, 189, 317] of the order lV(H2). For the quadrupole-induced 0223 and 2023 components of H2-H2, at the temperature of 40 K, such results differ from the exact zeroth, first and second moments by -10%, -10%, and +30% respectively. For the leading overlap-induced 0221 and 2021 components, we get similarly +14%, +12%, and -56%. These numbers illustrate the significance of a quantum treatment of the hydrogen pair at low temperatures. At room temperature, the semiclassical and quantum moments of low order differ by a few percent at most. Quantum calculations of higher-order moments differ, however, more strongly from their classical counterparts. 

Z. Gburski, C. G. Gray, and D. E. Sullivan. Higher-order spectral moments in collision-induced absorption Inert gas mixtures. Chem. 

Another approach to estimating spectral densities, which has the advantage of guaranteeing that the approximate functions are positive, can be based on the error bounds constructed in Section III-A for the spectral density broadened by a Lorentzian slit function. If we had a sufficient number of moments to make the error bounds very precise, then we could reduce the broadening as much as we like, so that the broadened distribution of spectral density becomes as close as we like to the true distribution. In order to estimate these higher moments, we should need to take advantage of some special feature of the distribution. For example, in the case of the harmonic vibrations of a crystalline solid, the distribution of frequencies lies between limits — co,nax and +comax, and is zero outside... 

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