Thermal correlation times

Figure 2 shows the thermal correlation times rc, calculated from the longitudinal relaxation times in the usual way (1). The time constant rc is a direct measure of the molecular mobility, and one recognizes that at room temperature the molecular mobility of cyclohexane is about 200 times greater than that of benzene. As the corresponding heats of adsorption (5) differ only by a factor 1.5, proton relaxation is extremely sensitive to specific interaction (c/. (6)). 

The reciprocal of the frequency can be defined as the rotational correlation time, tc, which represents the approximate time required for reorientation by random thermal tumbling. For values of tc < 5 x 10"9 s, an approximate theoretical solution for the linewidths of the three nitroxide ESR lines is given by (17) ... 

Figure 2.4 Carboxylic acid dimer in the potential energy minima with local vibrational states. OV represents the correlation time for a thermally activated proton transfer, and TU, the correlation time for tunneling transfer. (Reproduced with permission from ref. 29.)...

The single particle auto-correlation time tc in Eq. 9 can, of course, exhibit also a non-critical temperature dependence. Consider a set of independent hydrogen bonds with symmetric double well potentials and a barrier a between the wells. In this case the motion is thermally activated and tc shows an Arrhenius behaviour ... 

Figure 2. Temperature dependence of the correlation times for thermal motion as calculated from Figure 1...

The proton NMRD profile of an ethyleneglycol solution containing GdCb is reported in Fig. 5.53 at two different temperatures. The correlation time for electron relaxation, xv, is longer than in water. This could indicate that collisions of solvent molecules with the ion are slowed down in viscous solvents. r5o, which is related to the magnitude of the instantaneous ZFS induced by collisions, instead, does not change much. Therefore, the decrease of thermal motion as well as the... 

Two-dimensional (2D) spectroscopy is used to obtain some kind of correlation between two nuclear spins 7 and J, for instance through scalar or dipolar connectivities, or to improve resolution in crowded regions of spectra. The parameters to obtain 2D spectra are nowadays well optimized for paramagnetic molecules, and useful information is obtained as long as the conditions dictated by the correlation time for the electron-nucleus interaction are not too severe. Sometimes care has to be taken to avoid that the fast return to thermal equilibrium of nuclei wipes out the effects of the intemuclear interactions that are sought through 2D spectroscopy. 

This section is devoted to biological order, organization, and evolution. We have already seen in Appendix F that constructive integration of quantum and thermal correlations under appropriate conditions lead to a so-called CDS, i.e., an optimal spatiotemporal structure formed by the precise relations between time, size, and temperature scales. CDS suggests microscopic selforganization including Godel-like self-referential traits. 

Force field validation. In addition to ensuring that the force field reproduces results of QC calculations we have compared predictions of MD simulations using this force field with the available experimental data. Gas phase MD simulations using the quantum-chemistry based force field accurately reproduced the gas phase structure of DMNA as determined from electron diffraction studies. Liquid phase MD simulations of DMNA predicted the densities and solubility parameter as well as the activation energy and correlation times associated with molecular reorientation that are in good agreement with experimental data [34], As we will show in Section 4, comparison to structural and thermal data for the three pure crystalline polymorphs of HMX support the overall validity of our formulation and parameterization. 

This local field fluctuates with a correlation time T, about a nonvanishing average arising from the unequal populations of the electronic levels in thermal equilibrium. This average field is designated the susceptibility field, AB, where ... 

This review illustrates the above delineated characteristics of electron-transfer activated reactions by analyzing some representative thermal and photoinduced organometallic reactions. Kinetic studies of thermal reactions, time-resolved spectroscopic studies of photoinduced reactions, and free-energy correlations are presented to underscore the unifying role of ion-radical intermediates [29] in—at first glance—unrelated reactions such as additions, insertions, eliminations, redox reactions, etc. (Photoinduced electron-transfer reactions of metal porphyrin and polypyridine complexes are not included here since they are reviewed separately in Chapters 2.2.16 and 2.2.17, respectively.)... 

The phenomenon of EXAFS has been known for a considerable time (see ref. 126) but it has been applied to obtain structural information within the last decade only. From equation (2) it is seen that neighbour separation depends on the phase of the EXAFS oscillations, while the co-ordination number Nj and thermal correlation factor (Tj depend on the signal amplitude. In 1971 it was shown by Sayers, Stern, and Lytle that an appropriate Fourier analysis of the data gives a radial structure factor (j) R) from which one can locate the positions of the atoms surrounding the atom which absorbs the X-ray photon (for detailed discussion see refs. 123, 128—130). A second method of data analysis, involving curve fitting techniques, has been used also. ... 

Recent simulations by Straub et al. show that the non-Markovian theory of Section V may break down for very large friction and very large correlation times of the thermal bath. This failure is due to the fact that in such extreme (and unphysical) limits of the parameters, the well motion may become again the rate-limiting step in the process, in contrast to the theoretical assumption. - ... 

The Nb chemical shift in (775-C5H5)Nb(CO)4 in tetrahydrofuran had a thermal sensitivity measured as 0.38 ppmK from 210 to 340 K, " and the same compound in CH2CI2 a value of 0.42 ppmK . A linear decrease in shielding of the Nb nucleus, with increased temperature, of 0.25 ppmK for [CpNb(H)(CO)3] and [CpNb(D)(CO)3] has been measured, and also a non-linear decrease in linewidths with increased temperature for both of these compounds, indicative of increased molecular correlation times. [Et4N][Nb (CO)6] in THF showed a linear temperature dependence with a sensitivity of 0.18 ppmK over a range 203 to 323 K, and an increase in linewidth from 9.8 Hz (323 K) to 23 Hz (203 K). -... 

Fig. 15. Schematic of polar and alignment disorder as measured by XRD and NMR. Upper line polar disorder with random up or down orientations of the direction of the molecules with equal probabilities to restore the inversion symmetry for X-ray, the two directions are possible with equal probabilities on each site NMR cannot measure polar disorder, all sites are equivalent. Lower line alignment disorder of the molecules characterized by a long correlation time as compared to the inverse Larmor frequency coL, and librational disorder represented by ellipses of thermal-induced rotations (with angular amplitude possibly larger than the corresponding disalignment) around the mean alignment direction of the molecules. The frequencies of librations 1 jx is much larger than the Larmor frequency. For X-ray, both orientational disorders are mixed up with a preponderant contribution of the high-frequency librations for NMR, only alignment disorder remains.

---