The correlation time

Relaxation measurements provide a wealth of information both on the extent of the interaction between the resonating nuclei and the unpaired electrons, and on the time dependence of the parameters associated with the interaction. Whereas the dipolar coupling depends on the electron-nucleus distance, and therefore contains structural information, the contact contribution is related to the unpaired spin density on the various resonating nuclei and therefore to the topology (through chemical bonds) and the overall electronic structure of the molecule. The time-dependent phenomena associated with electron-nucleus interactions are related to the molecular system, and to the lifetimes of different chemical situations, for the resonating nucleus. Obtaining either structural or dynamic information, however, is only possible if an in-depth analysis of a series of experimental results provides sufficient data to characterize the system within the theoretical framework discussed in this chapter. 

Electrons switch between levels characterized by Ms values. Let us examine now an ensemble of n molecules, each with an unpaired electron, in a magnetic field at a given temperature. The bulk system is at constant energy but at the molecular level electrons move, molecules rotate, there are concerted atomic motions (vibrations) within the molecules and, in solution, molecular collisions. Is it possible to have information on these dynamics on a system which is at equilibrium The answer is yes, through the correlation function. The correlation function is a product of the value of any time-dependent property at time zero with the value at time t, summed up to a large number n of particles. It is a function of time. In this case the property can be the Ms value of an unpaired electron and the particles are the molecules. The correlation function has its maximum value at t = 0 since each molecule has one unpaired electron, the product of the 

Ms value at time zero times the Ms value at time t (t = 0) is either A A or — h (—Vi), i.e. always equal to + A. With time, some spins will change their orientation (Fig. 3.1 A), so some particles will contribute by — A. The value of the correlation function C(t) decreases. It can be shown that the decay of the correlation function can be approximated by an exponential in the absence of constraints (see Fig. 3.2)1. 

From Eq. (3.1), the correlation time xc is defined as the time constant for which the correlation function exponentially decays to zero. At a time small compared with rc, exp(—t/xc) — 1 and we expect that essentially all spins maintain their original value. Therefore, most of the products will be A. At times long with respect to rc, we expect that all spins have changed their orientations many times, so that on the average half of the spins will result with the same Ms with respect to their original values, and the other half will have the other Ms value. Under these conditions, statistically half of the products will be A and half — A, and the summation over a large number n of spins will yield zero. 

Strictly speaking, the time dependence of C(t) is not exponential for f zc by referring to the above example, after a time much shorter than rc but still larger than zero, it is likely that C t) = C(0). Indeed, an expansion of the initial part of C(t) would show a flat region around zero. In other words, it must be that dC(0)/df — 0, whereas for a true exponential dC(0)/df = l/4rc. Only for longer times does C(t) become effectively exponential. The small initial deviation 

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We call the correlation time it is equal to 1/6 Dj, where Dj is the rotational diffusion coefficient. The correlation time increases with increasing molecular size and with increasing solvent viscosity, equation Bl.13.11 and equation B 1.13.12 describe the rotational Brownian motion of a rigid sphere in a continuous and isotropic medium. With the Lorentzian spectral densities of equation B 1.13.12. it is simple to calculate the relevant transition probabilities. In this way, we can use e.g. equation B 1.13.5 to obtain for a carbon-13... 

Small molecules in low viscosity solutions have, typically, rotational correlation times of a few tens of picoseconds, which means that the extreme narrowing conditions usually prevail. As a consequence, the interpretation of certain relaxation parameters, such as carbon-13 and NOE for proton-bearing carbons, is very simple. Basically, tlie DCC for a directly bonded CH pair can be assumed to be known and the experiments yield a value of the correlation time, t. One interesting application of the measurement of is to follow its variation with the site in the molecule (motional anisotropy), with temperature (the correlation... 

However, in many applications the essential space cannot be reduced to only one degree of freedom, and the statistics of the force fluctuation or of the spatial distribution may appear to be too poor to allow for an accurate determination of a multidimensional potential of mean force. An example is the potential of mean force between two ions in aqueous solution the momentaneous forces are two orders of magnitude larger than their average which means that an error of 1% in the average requires a simulation length of 10 times the correlation time of the fluctuating force. This is in practice prohibitive. The errors do not result from incorrect force fields, but they are of a statistical nature even an exact force field would not suffice. 

In spin relaxation theory (see, e.g., Zweers and Brom[1977]) this quantity is equal to the correlation time of two-level Zeeman system (r,). The states A and E have total spins of protons f and 2, respectively. The diagram of Zeeman splitting of the lowest tunneling AE octet n = 0 is shown in fig. 51. Since the spin wavefunction belongs to the same symmetry group as that of the hindered rotation, the spin and rotational states are fully correlated, and the transitions observed in the NMR spectra Am = + 1 and Am = 2 include, aside from the Zeeman frequencies, sidebands shifted by A. The special technique of dipole-dipole driven low-field NMR in the time and frequency domain [Weitenkamp et al. 1983 Clough et al. 1985] has allowed one to detect these sidebands directly. 

In Eq. (4-62) Wq is the Larmor precessional frequency, and Tc is the correlation time, a measure of the rate of molecular motion. The reciprocal of the correlation time is a frequency, and 1/Tc may receive additive contributions from several sources, in particular I/t, where t, is the rotational correlation time, t, is, approximately, the time taken for the molecule to rotate through one radian. Only a rigid molecule is characterized by a single correlation time, and the value of Tc for different atoms or groups in a complex molecule may provide interesting chemical information. 

Figure 4-7. Schematic dependence (log-log plot) otTi and as functions of To, the correlation time. The minimum in Ti occurs at t = l/ojo.

Turning from chemical exchange to nuclear relaxation time measurements, the field of NMR offers many good examples of chemical information from T, measurements. Recall from Fig. 4-7 that Ti is reciprocally related to Tc, the correlation time, for high-frequency relaxation modes. For small- to medium-size molecules in the liquid phase, T, lies to the left side of the minimum in Fig. 4-7. A larger value of T, is, therefore, associated with a smaller Tc, hence, with a more rapid rate of molecular motion. It is possible to measure Ti for individual carbon atoms in a molecule, and such results provide detailed information on the local motion of atoms or groups of atoms. Levy and Nelson " have reviewed these observations. A few examples are shown here. T, values (in seconds) are noted for individual carbon atoms. 

Just as in phase transitions in statistical mechanical systems, observable quantities in PCA systems display singularities obeying simple power laws with universal critical exponents at the transition point. For example, letting ni be the number of sites with correlation length, and t be the correlation time, Kinzel [kinz85b] finds that for p ... 

Now, using the fact that the correlation time t is expected to scale asymptotically for large N as... 

In Fig. 7 actual numbers for the correlation times of the motion are included that apply for 2H NMR of deuterons in C—H bonds. It is evident that the line shape analysis of deuteron spectra should, in principle, provide a means to determine accurate values for the correlation times in a range of at least three orders of magnitude, the limits... 

Fig. 8. Calculated solid echo 2H NMR powder spectra for jumps between two sites related by the tetrahedral angle for ij =0, i.e. true absorption spectrum and Tj = 200 ps. xc is the correlation time of motion. R is the reduction factor, giving the total normalized intensity of the spectra for x, = 200 ps. (For x, = 0 all the spectra have total intensity 1)...

In order to find the correlation time ze = zp of rotational energy, it is necessary to eliminate the constant component Ke(00) from Eq. (1.56). 

However, only the left-hand side of the inequality has a clear, although qualitative, physical meaning. As far as collision time tc is concerned, its evaluation as p/ v) in Eq. (1.58) is rather arbitrary. Alternatively, it may be defined as the correlation time of the collisional processes which modulate the rotation. Using the mechanical equation of motion... 

As far as indirect methods are concerned (for instance, that of magnetic resonance), they measure solely the correlation times of orientational relaxation, which are integral characteristics of the process ... 

This connection between the correlation time of perturbation and that of response is a very general result independent of a model of molecular motion. It is valid not only when a molecule is perturbed by a sequence of instantaneous collisions (as in a gas), but also when it is subjected to perturbations that are continuous in time (caused by the nearest... 

Here te, tc are the correlation times of rotational and vibrational frequency shifts. The isotropic scattering spectrum corresponding to Eq. (3.15) is the Lorentzian line of width Acoi/2 = w0 + ydp- Its maximum is shifted from the vibrational transition frequency by the quantity coq due to the collapse of rotational structure and by the quantity A due to the displacement of the vibrational levels in a medium. 

It should be noted that there is a considerable difference between rotational structure narrowing caused by pressure and that caused by motional averaging of an adiabatically broadened spectrum [158, 159]. In the limiting case of fast motion, both of them are described by perturbation theory, thus, both widths in Eq. (3.16) and Eq (3.17) are expressed as a product of the frequency dispersion and the correlation time. However, the dispersion of the rotational structure (3.7) defined by intramolecular interaction is independent of the medium density, while the dispersion of the vibrational frequency shift (5 12) in (3.21) is linear in gas density. In principle, correlation times of the frequency modulation are also different. In the first case, it is the free rotation time te that is reduced as the medium density increases, and in the second case, it is the time of collision tc p/ v) that remains unchanged. As the density increases, the rotational contribution to the width decreases due to the reduction of t , while the vibrational contribution increases due to the dispersion growth. In nitrogen, they are of comparable magnitude after the initial (static) spectrum has become ten times narrower. At 77 K the rotational relaxation contribution is no less than 20% of the observed Q-branch width. If the rest of the contribution is entirely determined by... 

Inequality (6.67) is the softest criterion of perturbation theory. Its physical meaning is straightforward the reorientation angle (2.30) should be small. Otherwise, a complete circle may be accomplished during the correlation time of angular momentum and the rotation may be considered to be quasi-free. Diffusional theory should not be extended to this situation. When it was nevertheless done [268], the results turned out to be qualitatively incorrect orientational relaxation time 19,2 remained finite for xj —> 00. In reality t0j2 tends to infinity in this limit [27, 269]. 

A related experiment TOCSY (Total Correlation Spectroscopy) gives similar information and is relatively more sensitive than the REIAY. On the other hand, intensity of cross peak in a NOESY spectrum with a short mixing time is a measure of internuclear distance (less than 4A). It depends on the correlation time and varies as . It is positive for small molecules with short correlation time (o r <<1) and is negative for macromolecules with long correlation time (wr >>l) and goes through zero for molecules with 1 Relaxation effects should be taken into consideration for quantitative interpretation of NOE intensities, however. 

The second separation method involves n.O.e. experiments in combination with non-selective relaxation-rate measurements. One example concerns the orientation of the anomeric hydroxyl group of molecule 2 in Me2SO solution. By measuring nonselective spin-lattice relaxation-rat s and n.0.e. values for OH-1, H-1, H-2, H-3, and H-4, and solving the system of Eq. 13, the various py values were calculated. Using these and the correlation time, t, obtained by C relaxation measurements, the various interproton distances were calculated. The distances between the ring protons of 2, as well as the computer-simulated values for the H-l,OH and H-2,OH distances was commensurate with a dihedral angle of 60 30° for the H-l-C-l-OH array, as had also been deduced by the deuterium-substitution method mentioned earlier. 

Figure 4.9 Correlation time of the fluorescence anisotropy decay for the PMMA brush. The open and closed circles indicate the correlation times for the brush in acetonitrile and benzene, respectively.

The Mossbauer spectra of the complex [Fe(acpa)2]PF6 shown in Fig. 26 have also been interpreted on the basis of a relaxation mechanism [168]. For the calculations, the formalism using the modified Bloch equations again was employed. The resulting correlation times x = XlXh/(tl + Xh) are temperature dependent and span the range between 1.9 x 10 s at 110 K and 0.34 x 10 s at 285 K. Again the correlation times are reasonable only at low temperatures, whereas around 200 K increase of the population of the state contributes to... 

Locahzed motion can also lead to local variations in correlation times. Folded peptides with unfolded C- or N-terminal residues, for example, will have varying correlation times for the rigid and flexible parts of the molecule, resulting in different cross-relaxation rates. Such effects can usually be distinguished by the Unewidths and intensities of the corresponding diagonal signals, since the autorelaxation rates also depend on the correlation time. 

Exchange-transferred spectroscopy was introduced with the finding of the etNOE [97] and its theoretical explanation in terms of fast exchange several years later [98] laid the basis for the large variety of applications being present today. The core element of etNOE is the dependence of the cross-relaxation rate 0 ° on the correlation time T,-. The overall cross-relaxation rate is defined by ... 

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