Reorientational correlation time

Models for description of liquids should provide us with an understanding of the dynamic behavior of the molecules, and thus of the routes of chemical reactions in the liquids. While it is often relatively easy to describe the molecular structure and dynamics of the gaseous or the solid state, this is not true for the liquid state. Molecules in liquids can perform vibrations, rotations, and translations. A successful model often used for the description of molecular rotational processes in liquids is the rotational diffusion model, in which it is assumed that the molecules rotate by small angular steps about the molecular rotation axes. One quantity to describe the rotational speed of molecules is the reorientational correlation time T, which is a measure for the average time elapsed when a molecule has rotated through an angle of the order of 1 radian, or approximately 60°. It is indirectly proportional to the velocity of rotational motion. 

Usually, nuclear relaxation data for the study of reorientational motions of molecules and molecular segments are obtained for non-viscous liquids in the extreme narrowing region where the product of the resonance frequency and the reorientational correlation time is much less than unity [1, 3, 5]. The dipolar spin-lattice relaxation rate of nucleus i is then directly proportional to the reorientational correlation time p... 

Table 4.5-1 gives values for the fit parameters and the reorientational correlation times calculated from the dipolar relaxation rates. 

Table 4.5-1 Reorientational correlation times rat 357 l< and fit parameters activation energy E, ...

As an example of tetra-coordinate cobalt(II) systems, the NMRD profile of cobalt(II)-substituted carbonic anhydrase (MW 30,000) at high pH is reported (Fig. 14). The metal ion is coordinated to three histidines and to a hydroxide ion (48). The NMRD profile shows a cos Cg dispersion centered around 10 MHz, which qualitatively sets the correlation time around 10 s. As the reorientational correlation time of the molecule is much longer, this value is a measure of the effective electronic relaxation time. A quantitative... 

Chromium(III) has a ground state in pseudo-octahedral symmetry. The absence of low-lying excited states excludes fast electron relaxation, which is in fact of the order of 10 -10 ° s. The main electron relaxation mechanism is ascribed to the modulation of transient ZFS. Figure 18 shows the NMRD profiles of hexaaqua chromium(III) at different temperatures (62). The position of the first dispersion, in the 333 K profile, indicates a correlation time of 5 X 10 ° s. Since it is too long to be the reorientational time and too fast to be the water proton lifetime, it must correspond to the electron relaxation time, and such a dispersion must be due to contact relaxation. The high field dispersion is the oos dispersion due to dipolar relaxation, modulated by the reorientational correlation time = 3 x 10 s. According to the Stokes-Einstein law, increases with decreasing temperature, and... 

The reorientational correlation time can be predicted for spherical rigid particles, according to the Stokes Einstein equation (75-77) ... 

In the previous discussion, the electron-nucleus spin system was assumed to be rigidly held within a molecule isotropically rotating in solution. If the molecule cannot be treated as a rigid sphere, its motion is in general anisotropic, and three or five different reorientational correlation times have to be considered 79). Furthermore, it was calculated that free rotation of water protons about the metal ion-oxygen bond decreases the proton relaxation time in aqua ions of about 20% 79). A general treatment for considering the presence of internal motions faster than the reorientational correlation time of the whole molecule is the Lipari Szabo model free treatment 80). Relaxation is calculated as the sum of two terms 8J), of the type... 

When the temperature is lowered and/or the viscosity of the solution is increased by using glycerol-water mixtures as solvent, the reorientational correlation time increases. Since the reorientational time is the correlation time for nuclear relaxation, the effects on the NMRD profile (Pig. 27) are (i) higher relaxivity values at low frequencies (ii) a shift toward lower... 

The NMRD profiles of V0(H20)5 at different temperatures are shown in Fig. 35 (58). As already seen in Section I.C.6, the first dispersion is ascribed to the contact relaxation, and is in accordance with an electron relaxation time of about 5 x 10 ° s, and the second to the dipolar relaxation, in accordance with a reorientational correlation time of about 5 x 10 s. A significant contribution for contact relaxation is actually expected because the unpaired electron occupies a orbital, which has the correct symmetry for directly overlapping the fully occupied water molecular orbitals of a type (87). The analysis was performed considering that the four water molecules in the equatorial plane are strongly coordinated, whereas the fifth axial water is weakly coordinated and exchanges much faster than the former. The fit indicates a distance of 2.6 A from the paramagnetic center for the protons in the equatorial plane, and of 2.9 A for those of the axial water, and a constant of contact interaction for the equatorial water molecules equal to 2.1 MHz. With increasing temperature, the measurements indicate that the electron relaxation time increases, whereas the reorientational time decreases. 

In Eqs. (4)-(7) S is the electron spin quantum number, jh the proton nuclear magnetogyric ratio, g and p the electronic g factor and Bohr magneton, respectively. r//is the distance between the metal ion and the protons of the coordinated water molecules, (Oh and cos the proton and electron Larmor frequencies, respectively, and Xr is the reorientational correlation time. The longitudinal and transverse electron spin relaxation times, Tig and T2g, are frequency dependent according to Eqs. (6) and (7), and characterized by the correlation time of the modulation of the zero-field splitting (x ) and the mean-square zero-field-splitting energy (A. The limits and the approximations inherent to the equations above are discussed in detail in the previous two chapters. 

The molecular reorientational correlation time tends to dominate the overall correlation time of low molecular weight Gd(III) chelates, particularly in the high field region, and therefore represents a key parameter in governing their relaxivity. The effect of the increase in x on the shape and amplitude of the NMRD profiles was understood in detail early on and, as a consequence, the attempts at optimizing the relaxivity were primarily focused on slowing down the rotation by increasing the size of the... 

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-... 

The correlation time is the reorientational correlation time. This is generally the rotational correlation time unless internal motions of a group in a molecule are faster. In the latter case, pairs of nuclei in the same molecule can have different reorientational correlation times. 

In this last case, the ratio of enhancements of the longitudinal and transverse relaxation rates, relative to those of the "free" solvate cation in solution, gives access to the value of the reorientational correlation time of the cation interacting with the surface. 

Nmr methods have unrivalled potential to explore interfaces, as this account has striven to show. We have been able to determine the mobility of hydrated sodium cations at the interface of the Ecca Gum BP montmorillonite, as 8.2 ns. We have been able to measure the translational mobility of water molecules at the interface, their diffusion coefficient is 1.6 10 15 m2.s. We have been able to determine also the rotational mobility of these water adsorbate molecules, it is associated to a reorientational correlation time of 1.6 ns. Furthermore, we could show the switch in preferred reorientation with the nature of the interlayer counterions, these water molecules at the interface tumbling about either the hydrogen bond to the anionic surface or around the electrostatic bond to the metallic cation they bear on their back. And we have been able to achieve the orientation of the Ecca Gum BP tactoids in the strong magnetic field of the nmr spectometer. 

Principle possibility of measuring hydrodynamic radii of colloidal nanoparticles rT from the reorientation correlation times tr was demonstrated in [132] for three quasi-spherical heteropoly anions in buffered aqueous solutions at several temperatures. The analysis of EPR line widths of V4+ ions by Kivelson equations allowed to determine Tr, from which rt were estimated by the ratio ... 

The isotropic signal delivers (rotation-free) information on the temporal evolution of the population numbers of the investigated vibrational transition(s). The induced dichroism is governed by the time constant ror (second-order reorientational correlation time, 1 = 2) and possibly population redistribution that may contribute to the loss of induced optical anisotropy. The zero-setting of the delay time scale (maximum overlap between pump and probing pulses) is determined by a two-photon absorption technique in independent measurements with an accuracy of better than 0.2 ps (67). 

Probe site dynamics on a time scale dictated by properties of the nucleus in question typically nanoseconds (exchange dynamics, molecular reorientational correlation times) is available. 

Figure 10.4 Schematic diagram of the three types of relaxation mechanisms and the major parameters in the relaxation, which are the number of coordinated water molecules, water exchange rate (k ), and reorientational correlation time (xr).

The second-sphere relaxation is the motion of the water molecules that are hydrogen bonded to the metal center. These water molecules have a longer residence time around the paramagnetic center with respect to the outer-sphere water molecules. Three parameters of relaxation mechanism that may increase the relaxivity have been intensely studied, namely, the number of coordinated water molecules (q), the water exchange rate (kex = 1/tm) in Equation 10.3, and the reorientational correlation time (tr) in Equation 10.7. 

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