Exchange correlation time

The movements capable of relaxing the nuclear spin that are of interest here are related to the presence of unpaired electrons, as has been discussed in Section 3.1. They are electron spin relaxation, molecular rotation, and chemical exchange. These correlation times are indicated as rs (electronic relaxation correlation time), xr (rotational correlation time), and xm (exchange correlation time). All of them can modulate the dipolar coupling energy and therefore can cause nuclear relaxation. Each of them contributes to the decay of the correlation function. If these movements are independent of one another, then the correlation function decays according to the product... 

In these equations, the symbols have their customary meanings (see Toth et al. in this volume for an excellent review of the topic), and the correlation times given in Eq. (3) have the following typical values at 50 MHz in water Tle (electron spin-lattice relaxation time) =10 ns, T2e (electron spin-spin relaxation time) = 1 ns, rm (inner sphere water exchange correlation time) = 130 ns [3], and rR = 60 ps. These values, in the context of Eq. (1 - 3), show why rotational dynamics control relaxivity for such chelates. 

Shimiza et al. have used Zn linewidths to investigate the interaction of zinc with apothermolysin and with imidazoles and carboxylate anions. While giving interesting qualitative results, there are considerable quantitative uncertainties as the rates of exchange, correlation times and asymmetries of the zinc sites are unknown. 

We will rely on the so-called adiabatic local density approximation (ALDA) to describe the exchange-correlation time-dependent functional Vxc[n, t), based on the exchange-correlation of a free electron-gas [25-28]. 

It should be noted that long electron spin-lattice relaxation times (Ti 10" to 10 s) impede recording of the NMR spectra of radicals. Therefore lines in the NMR spectra of radicals become observable only at high rates of electron spin exchange. Spin exchange correlation time Tex is directly proportional to the viscosity rj and inversely proportional to the temperature and radical concentration [R] rj/T[R ]. Therefore,... 

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. 

In presence of molecular motion the NMR line shape will change. A particularly simple situation arises, if the motion is rapid on timescale defined by the inverse width of the spectrum in absence of motion 6 1. In this fast exchange limit, which in 2H NMR is reached for correlation times tc < 1CT7 s, the motion leads to a partially averaged quadrupole coupling and valuable information about the type of motion can directly be obtained from analysis of the resulting line shapes. The NMR frequency is then given by... 

Fig. 7. Theoretical line shapes resulting from an interchange between two NMR frequencies

Fig. 13. Calculated 2H solid echo spectra for log-Gaussian distributions of correlation times of different widths. Note the differences of the line shapes for fully relaxed and partially relaxed spectra. The centre of the distribution of correlation times is given as a normalized exchange rate a0 = 1/3tc. For deuterons in aliphatic C—H bonds the conversion factor is approximately 4.10s sec-1...

Fig. 29. Observed and calculated 2H NMR spectra for the mesogenic groups of a) the nematic (m = 2), b) the smectic (m = 6) liquid crystalline polymer in the glassy state, showing the line shape changes due to the freezing of the jump motion of the labelled phenyl ring. The exchange frequency corresponds to the centre of the distribution of correlation times. Note that the order parameters are different, S = 0.65 in the frozen nematic, and S = 0.85 in the frozen smectic system, respectively...

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... 

Note in particular that the exchange-correlation functional that emCTges here does not involve the kinetic energy. From the perspective of the DFT literature, (3.16) is a formulation of the Hohenberg-Kohn functional that is constructed to ensure that the functional derivatives required for variational minimization actually exist. We return to these issues in Sect. 3.3. Also note that in the time-dependent case the external potential V(r, )is often considered to be explicitly... 

All these methods have in common that the receptor is not detected so that no size limit applies. Relaxation-based exchange-transferred experiments actually show best performance for longest correlation times, i.e. very large receptors. 

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 ... 

As described in Section 9.2.4.5, CCR can be used to obtain dihedral angular information. Since CCR processes depend linearly on the correlation time, the requirements for exchange-transferred spectroscopy are fulfilled similarly to the etNOE. 

Note that in all current implementations of TDDFT the so-called adiabatic approximation is employed. Here, the time-dependent exchange-correlation potential that occurs in the corresponding time-dependent Kohn-Sham equations and which is rigorously defined as the functional derivative of the exchange-correlation action Axc[p] with respect to the time-dependent electron-density is approximated as the functional derivative of the standard, time-independent Exc with respect to the charge density at time t, i. e.,... 

It is clear from the above equations that numerous parameters (proton exchange rate, kcx = l/rm rotational correlation time, tr electronic relaxation times, 1 /rlj2e Gd proton distance, rGdH hydration number, q) all influence the inner-sphere proton relaxivity. Simulated proton relaxivity curves, like that in Figure 3, are often used to visualize better the effect of the... 

Figure 3 Effect of the water exchange rate, kex, and the rotational correlation time, rR, on inner-sphere proton relaxivity. The plot was simulated for a particular value of the longitudinal electron spin relaxation rate, 1/Tie — 5.28xlOss 1. The marketed contrast agents all have relaxivities around 4—5mM 1s 1 in contrast to the theoretically attainable values over lOOrnM-1 s 1, and this is mainly due to their fast rotation...

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