Angular correlation function, decay

The complex nature of the slow mode responsible for the long-time behavior of first rank correlation functions for a first rank interaction potential is illustrated by the composition of the eigenvector corresponding to the slow mode 11a in Table XI, for Uj = 3 and o) = 0.5. Note that n 1, tij, ii, J2 describe the magnitudes and the orientations of the momentum vectors Lj and L2 j is referred to the orientation of L, -t- Lj, 7, and J2 are related to the orientations of the two bodies, and the total orientational angular operator defines the quantum number J finally J, which is not included in this table, is the total angular momentum quantum number, and it is always equal to 1 for first rank orientational and momentum correlation functions, and to 2 for second rank correlation functions. In Fig. 11 we show the first rank correlation functions for different collision frequencies of body 1. The second rank correlation function decays are plotted in Fig. 12. The librational motions in the wells are more important than they were in the first rank potential case (since there is now a more accentuated curvature of the potential wells). 

Fig. 18. Decay scheme of In and the perturbed angular correlation functions W(0) of In-labeled compounds...

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. 

Reorientation dynamics in liquids is described by either diffusion constants, D., or reorientational correlation times, since these two parameters are closely correlated. D. is the diffusion rate about a given molecular axis while is the time period required for the angular correlation function to decay to 1/e of its initial value [34,35]. For symmetric-top molecules, such as two diffusion constants, and D, are usually required to characterize the overall motion. and represent rotational diffusion about and of the top axis, respectively. The overall motion is now characterized by an effective reorientational correlation time, that, in the limit of small-step diffusion, is given by [36]... 

Another tractable case is that of relatively rapid internal rotations or librations superimposed on a slower overall molecular tumbling. The presence of such motions is revealed by unexpected NOE and Tj s, and by a nonstandard variation of with spectrometer frequency. In this case also the angular correlation function does not decay exponentially. Instead, it decays at a rate determined by the internal motion to some fraction of its original value. This fraction quantitatively represents the mean angular constraint on the internal motion, in that it is the angular correlation that remains when the internal motion alone has completed its randomizing work. G x) then continues to decay more slowly to zero at a rate (here assumed to be much slower) determined by the overall molecular tumbling. 

Judging by these results the angular momentum relaxation in a dense medium has the form of damped oscillations of frequency jRo = (Rctc/to)i and decay decrement 1/(2tc). This conclusion is quantitatively verified by computer experiments [45, 54, 55]. Most of them were concerned with calculations of the autocorrelation function of the translational velocity v(t). However the relation between v(t) and the force F t) acting during collisions is the same as that between e> = J/I and M. Therefore, the results are qualitatively similar. In Fig. 1.8 we show the correlation functions of the velocity and force for the liquid state density. Oscillations are clearly seen, which point to a regular character of collisions and non-Markovian nature of velocity changes. 

This latter expression has been used to simplify KD(t)- Note that the time dependences of the linear and angular momentum autocorrelation functions depend only on interactions between a molecule and its surroundings. In the absence of torques and forces these functions are unity for all time and their memories are zero. There is some justification then for viewing these particular memory functions as representing a molecule s temporal memory of its interactions. However, in the case of the dipolar correlation function, this interpretation is not so readily apparent. That is, both the dipolar autocorrelation function and its memory will decay in the absence of external torques. This decay is only due to the fact that there is a distribution of rotational frequencies, co, for each molecule in the gas phase. In... 

Figure 18. Computer simulation of the effects of excitation on the decay of the angular velocity u. Curve 1 denotes the equilibrium correlation function Curves 2... 

Fig. 4. Time dipole correlation functions C(t) of water in critical state (left top), in bulk liquid water at 30°C (left center), in a monolayer on fluorophlogopite mica (left bottom), in LTA bonded to the first 4 Na+ ions (right top), in SB A-15 heated to 300°C for 2 hrs (right center), and in fully hydrated SBA-15 (right bottom). The normalized total correlation functions, obtained according to Eq. (9) involve vibrations of the transition dipole of the (v+5) band displayed as rapid oscillations. Rotational correlations including angular perturbations appear as envelopes of the vibrational correlation functions. The inertial rotational motion about the least rotational axis of the water molecule is indicated as a quadratic decay C(t) - (kT/I) t2 at times 0 - 0.05 psec in each C(t) vs. t graph. The graphs on the left are reproduced from ref. 18.

K and followed the (T-T )"l relation, with T < 1.0 K, indicating the formation of local order in the liquid. While the presence of such order is detected by such Kerr-effect experiments, the individual terms in equations (23) and (24) are not easily extracted from the data. The theory of the dynamic Kerr-effect for an angularly correlated system appears to be extremely complicated, partly because of the difference between step-on and step-off responses discussed above. In the presence of angular correlations. Cole (86) showed that the induced dipole moment term gives rise and decay functions, for step-on and step-off fields, which are equivalent and give information on I 2) [Pg.266]

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. 

Now consider rotational diffusion of a rigid symmetric top in an oriented environment such as a lipid bilayer or other liquid crystal. The top (e.g., cylinder) could represent one of the lipids, a probe, or a protein. It may be useful to think of the vector wobbling in a cone , as sketched in Figure 5. The correlation function clearly will decay to a nonzero plateau value, because angular averaging is incomplete. One may also anticipate that the relaxation will be more rapid than free diffusion due to effects of the ordering potential. 

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