Angular momentum correlation functions

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

Using solution (1.37) in definition (1.4), one has the angular momentum correlation function... 

The qualitative difference between low-density and high-density rotational relaxation is clearly reflected in the Fourier transform of the normalized angular momentum correlation function ... 

Fig. 1.11. The normalized angular momentum correlation function Kj(t)/Kj(0) at k — 0.25 in differential (curve a), integral (curve b) and impact (curve c) theories.

Of course, knowledge of the entire spectrum does provide more information. If the shape of the wings of G (co) is established correctly, then not only the value of tj but also angular momentum correlation function Kj(t) may be determined. Thus, in order to obtain full information from the optical spectra of liquids, it is necessary to use their periphery as well as the central Lorentzian part of the spectrum. In terms of correlation functions this means that the initial non-exponential relaxation, which characterizes the system s behaviour during free rotation, is of no less importance than its long-time exponential behaviour. Therefore, we pay special attention to how dynamic effects may be taken into account in the theory of orientational relaxation. 

Without resorting to the impact approximation, perturbation theory is able to describe in the lowest order in both the dynamics of free rotation and its distortion produced by collisions. An additional advantage of the integral version of the theory is the simplicity of the relation following from Eq. (2.24) for the Laplace transforms of orientational and angular momentum correlation functions [107] ... 

Zatsepin V. M. To experimental determination of angular momentum correlation function in liquids, Ukrainian Phys. J. 21, 48-52 (1976). 

All of these studies likewise show clearly that in liquids with potentials that have a strong noncentral character there is an interval of time for which the angular momentum correlation function is negative (see Figures 12 and 13) whereas in liquids for which the pair potential has a small noncentral character this function remains positive and changes very little over the... 

Introduction of the normalized angular momentum correlation functions, Aj(t), into this integral, followed by an integration by parts yields... 

The mean square torque is taken from computer experiments. Nevertheless, it could have been found from the infrared bandshapes. Likewise the integral in this expression can be found from the experimental spin rotation relaxation time, or it can be found directly from the computer experiment as it is here. The memory function equation can be solved for this memory. The corresponding angular momentum correlation function has the same form as v /(0 in Eq. (302) with... 

Finally, consider the power spectra of the experimental approximate correlation functions which are displayed in Figures 24, 29, and 34. Note that each of these spectra has been normalized to unity at co = 0. Note also that the experimental spectrum from the angular momentum correlation function is much broader than the experimental velocity autocorrelation power spectra. The power spectra of the Gaussian II autocorrelation functions are in much better agreement with the experimental spectra at all frequencies than the power spectra of the other approximate autocorrelation functions. 

The important point to note here is that the 2nd moment of Ky(t) depends on the 2nd and 4th moments of y(t). The 2nd moments of each of the three previously mentioned autocorrelation functions may be calculated from ensemble averages of appropriate functions of the positions, velocities, and accelerations created in the dynamics calculations. Likewise, the 4th moment of the dipolar autocorrelation function may also be calculated in this manner. However the 4th moments of the velocity and angular momentum correlation functions depend on the derivative with respect to time of the force and torque acting on a molecule and, hence, cannot be evaluated directly from the primary dynamics information. Therefore, these moments must be calculated in another manner before Eq. (B.3) may be used. 

We have already seen that the assiimption of uncorrelated collisions leads to an exponential decay of the angular momentum correlation function. This is rarely appropriate for the fluids that have been simulated. If CQ)(t) has the wrong shape the leading term in the cumulant expansion is incorrect. We deduce that these models are unlikely to be more useful than straight diffusion in most circumstances. However there are some "gas-like" low torque fluids and it is useful to be able to compare the two models. 

In spite of their popularity among spectroscopists it seons unlikely that the use of these models to fit spectral data is justified unless one has good reason to believe that the angular momentum correlation functions decay approximately exponentially with no cage effect. 

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