Reorientational correlation functions

Dynamic information such as reorientational correlation functions and diffusion constants for the ions can readily be obtained. Collective properties such as viscosity can also be calculated in principle, but it is difficult to obtain accurate results in reasonable simulation times. Single-particle properties such as diffusion constants can be determined more easily from simulations. Figure 4.3-4 shows the mean square displacements of cations and anions in dimethylimidazolium chloride at 400 K. The rapid rise at short times is due to rattling of the ions in the cages of neighbors. The amplitude of this motion is about 0.5 A. After a few picoseconds the mean square displacement in all three directions is a linear function of time and the slope of this portion of the curve gives the diffusion constant. These diffusion constants are about a factor of 10 lower than those in normal molecular liquids at room temperature. 

Eagles T. E., McClung R. E. D. Reorientational correlation functions and memory functions in the. /-diffusion limit of the extended rotational diffusion model, Chem. Phys. Lett. 22, 414-18 (1973). 

Here the vector rj represents the centre of mass position, and D is usually averaged over several time origins to to improve statistics. Values for D can be resolved parallel and perpendicular to the director to give two components (D//, Dj ), and actual values are summarised for a range of studies in Table 3 of [45]. Most studies have found diffusion coefficients in the 10 m s range with the ratio D///Dj between 1.59 and 3.73 for calamitic liquid crystals. Yakovenko and co-workers have carried out a detailed study of the reorientational motion in the molecule PCH5 [101]. Their results show that conformational molecular flexibility plays an important role in the dynamics of the molecule. They also show that cage models can be used to fit the reorientational correlation functions of the molecule. 

A. Molecular Reorientation Correlation Function, Spectrum, and Susceptibility... 

R. M. Lynden-Bell and A. J. Stone, Reorientational correlation functions, quaternions and Wigner rotation matrices, Molec. Simul., 3 (1989), 271. 

We conclude this section with some brief comments about microscopic dynamics at liquid interfaces. Molecular dynamic simulations of the dynamic properties of liquid interfaces have been limited to the calculation of equilibrium time correlation functions. The methodology of these calculations has been discussed earlier. One property that has received much attention is the molecular reorientation correlation function. If e(r) is a unit vector fixed in the molecular frame, the nth order time correlation function is defined by... 

When a second rank potential ( 2 0) is considered, there are significant differences in behavior of both the reorientational correlation functions (as in the 2BSM) and in the momentum correlation functions. Tables XI and XII give the results for first and second rank correlation functions, respectively. In both cases we have at least three decay modes. 

We now apply a simple version of this formalism to a calculation of the reorientational correlation functions for the linear... 

Figure 3 illustrates this problem for V3 of acetonitrile which is only a weak Raman band with a long wing to low frequency. The resulting reorientational correlation functions (fig. 4) are very noisy and t values are imprecise. With photon counting or multichannel detection (Raman) or fast scanning (F.T. infrared) these problems can be overcome, but only at the expense of a longer experimental time. Base line determination is an even more... 

To study the reorientational motion associated with 1=1 and 2, we calculated the single-particle reorientational correlation functions defined by [25]... 

FIGURE 2.6 (Upper panel) Plot for the reorientational correlation function against time for a representative composition (Xj = 0.1). (Lower panel) Product of the translational diffusion coefficient Dj. and the average orientational correlation time x, of the first-rank correlation function as a function of composition. Note that the solid line and dashed line indicate the hydrodynamic predictions with the stick and slip boundary conditions, respectively. 

Therefore, the conditional probability is fully determined by solving the eigenvalue problem given in Eq. (7.17). This is required to evaluate the reorientational correlation functions because... 

For magnetic resonance relaxation (L = 2), the reorientational correlation functions [Eq. (7.20)] may be evaluated for a spherical molecule (n = n = 0) with the equilibrium probability distribution /(fio) = 1/47t. Since the molecules are assumed to reorient isotropically, all correlation functions decay exponentially with a single correlation time... 

The reorientational correlation functions can be evaluated by calculating the angles between vectors at specified offsets using the direct method. Here we present two equivalent representations in the Cartesian frame that are useful for both computational and theoretical reasons. From the x, y, and z projections of A, and definition of the Cartesian tensor, ... 

The reorientational correlation functions for asymmetric tops (all three eigenvalues distinct) have three exponential decays for = 1, and five for = 2. Their explicit form is given in Ref. 25. 

So far the comments on reorientational correlation functions have been general. This subsection describes the form of the decay for a particularly important model a rigid particle undergoing diffusive motion. The diffusive limit is not always applicable for liquids (small molecules and methyl groups can show significant inertial effects), but is generally observed for medium to large molecules in solution and is useful for many analyses. 

As described in depth in Ref. 4, the starting point for this treatment is the rotational diffusion equation of a rigid body, which can be elegantly solved by exploiting its correspondence with the time-dependent Schrodinger equation. The solution is an infinite series of spherical harmonics. Recalling that the reorientational correlation functions are expressible as spherical harmonics, the orthonormality condition leads to the... 

The simple solutions found for the isotropic case (see Section 4.2) no longer drop out of the diffusion equation, and the reorientational correlation functions are now written as... 

Figure 6 The t = I and 2 reorientational correlation functions for a tod undergoing restricted (solid) and free (dashed) rotational diffusion...

It is useful for Section 4.5 to sketch out qualitatively the form of a reorientational correlation function for a vector in a protein. The fast motion (libration and sometimes isomerization) is restricted, and hence is associated with an order parameter. The decay to the plateau value is complicated for even simple models, but for here assume a single exponential with a fast decay time Xf-. 

The time-dependent anisotropy, r(r), in a fluorescence depolarization experiment is directly related to the reorientational correlation function ... 

A useful and common way of describing the reorientation dynamics of molecules in the condensed phase is to use single molecule reorientation correlation functions. These will be described later when we discuss solute molecular reorientational dynamics. Indirect experimental probes of the reorientation dynamics of molecules in neat bulk liquids include techniques such as IR, Raman, and NMR spectroscopy. More direct probes involve a variety of time-resolved methods such as dielectric relaxation, time-resolved absorption and emission spectroscopy, and the optical Kerr effect. The basic idea of time-resolved spectroscopic techniques is that a short polarized laser pulse removes a subset of molecular orientations from the equifibrium orientational distribution. The relaxation of the perturbed distribution is monitored by the absorption of a second time-delayed pulse or by the time-dependent change in the fluorescence depolarization. 

A systematic study of the rotational relaxation, of the diatomic solute described above, at the water/CCl4 interface has been carried out." The solute molecules equilibrium reorientation correlation functions, Eq. [45], were evaluated at different interface locations and in the bulk of the two solvents. Some of the results for the reorientation time are reproduced in Figure 11. 

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