Momentum correlations

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 ... 

In the impact approximation (tc = 0) this equation is identical to Eq. (1.21), angular momentum relaxation is exponential at any times and t = tj. In the non-Markovian approach there is always a difference between asymptotic decay time t and angular momentum correlation time tj defined in Eq. (1.74). In integral (memory function) theory Rotc is equal to 1/t j whereas in differential theory it is 1/t. We shall see that the difference between non-Markovian theories is not only in times but also in long-time relaxation kinetics, especially in dense media. 

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, the effect of excluded volume is opposite and greatly exceeds that shown in Fig. 1.10, which is produced by uncorrelated collective interaction. Unfortunately, neither of them results in sign-alternating behaviour of angular or translational momentum correlation functions. This does not have a simple explanation either in gas-like or solid-like models of liquids. As is clearly seen from MD calculations, even in... 

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. 

Table 2.1. Angular momentum correlation times tj of various molecules in liquefied gases [93J.

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] ... 

In the present section the general kinetic equation (3.26) will be solved within the Keilson-Storer model for an arbitrary angular momentum correlation [163], We consider here the case of spherical molecules (for linear molecules see Appendix 5). The corresponding initial condition is the equilibrium distribution... 

The physical meaning of and f L.., is obvious they govern the relaxation of rotational energy and angular momentum, respectively. The former is also an operator of the spectral exchange between the components of the isotropic Raman Q-branch. So, equality (7.94a) holds, as the probability conservation law. In contrast, the second one, Eq. (7.94b), is wrong, because, after substitution into the definition of the angular momentum correlation time... 

Gillen K. T., Douglas D. S., Malmberg M. S., Maryott A. A. NMR relaxation study of liquid CCI3F. Reorientational and angular momentum correlation times and rotational diffusion, J. Chem. Phys. 57, 5170-9 (1972). 

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

From this expression we see that the friction cannot be determined from the infinite-time integral of the unprojected force correlation function but only from its plateau value if there is time scale separation between the force and momentum correlation functions decay times. The friction may also be estimated from the extrapolation of the long-time decay of the force autocorrelation function to t = 0, or from the decay rates of the momentum or force autocorrelation functions using the above formulas. 

Notice that this problem differs from one analyzed by Mazur and Montroll16 since they were interested in the time relaxed momentum correlation functions . In an equilibrium ensemble it is known that both the qj and the p have a Gaussian distribution,15 and the distribution of momenta is characterized by... 

The expression given in this equation is easier to analyze than the equivalent one for Pr, since it involves the momentum correlation function directly rather than the inverse of the correlation matrix. It is a simple matter now to extract information related to the decay of correlations starting from Eq. (44). We note, first of all, that the functions irs(0 can be expressed in terms of the Ar(t), and these have the property lim,., Ar(i) = 0. Let us also set... 

At the other extreme we might expect a fluid to have some characteristic of a simple Einstein solid, i.e., a collection of independent oscillators each oscillating at the same frequency %. The linear momentum correlation function and its memory would then simply be... 

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... 

A. A Simple Model for Linear and Angular Momentum Correlations... 

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. 

B. J. Berne, Linear and Angular Momentum Correlations in Liquids and the Memory Function, unpublished. 

Note that the angular momentum-momentum correlation function related to the vibrational-rotation friction too is highly nonexponential. This again could significantly alter the 2 dependence of the rate. However, for the systems studied here, the contribution of vibration-rotation coupling is negligible. 

Yi and Ys - gyromagnetic ratio of spin 1 and spin S nuclear spin, rJS = intemuclear distance, tr= rotational correlation time, x< = reorientation correlation time, xj = angular momentum correlation time, Cs = concentration of spin S, Cq = e2qzzQ/h = quadrupole coupling constant, qzz = the electric field gradient, Q = nuclear electric quadrupole moment in 10 24 cm2, Ceff = effective spin-rotational coupling constant, a = closest distance of appropriate of spin 1 and spin S, D = (DA+DB)/2 = mutual translational self diffusion coefficient of the molecules containing I and S, Ij = moment of inertia of the molecule, Ao = a// - ol-... 

Schinke, R. (1988c). Angular momentum correlation in the photodissociation of H2O2 at 193 nm, J. Phys. Chem. 92, 4015-4019. 

The more obvious and consistent deviations from the hard sphere theory occur, at the low density values, due to the effects of attractive forces in the real system. We can attempt to correct for these effects using a method described previously (27-30) for the analysis of angular momentum correlation times in supercritical CFjj and CFjj mixtures with argon and neon. We replace the hard sphere radial distribution function at contact hs with a function gp (0) which uses the more realistic... 

If one corrects for the effects of attractive forces, the corrected values are much smaller than the observed values the correction overestimates the importance of attractive forces on the value of the diffusion coefficient. Although attractive forces have an effect on the diffusion coefficient at low density, they are not nearly as Important as they are in determining the value of the angular momentum correlation time. Diffusion is primarily determined by the repulsive forces between molecules, even at the lowest densities. 

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