Redfield limit

The last assumption is very fundamental. It results in time-independent transition probabilities and makes a clean theory possible. It requires that the product of the time scale of the decay time for the tcf (called the correlation time and denoted x ) and the strength of the perturbation (in angular frequency units) has to be much smaller than unity (17-20). This range is sometimes denoted as the Redfield limit or the perturbation regime. 

The recent versions of the slow motion approach were applied to direct fitting of experimental data for a series of Ni(II) complexes of varying symmetry (97). An example of an experimental data set and a fitted curve is shown in Fig. 9. Another application of the slow-motion approach is to provide benchmark calculations against which more approximate theoretical tools can be tested. As an example of work of this kind, we wish to mention the paper by Kowalewski et al. (98), studying the electron spin relaxation effects in the vicinity and beyond the Redfield limit. 

Westlund developed also a theory for PRE in the ZFS-dominated limit for S = 1, which included a stringent Redfield-limit approach to the electron spin relaxation in this regime (118). Equations (35) and (38) were used as the starting point also in this case. Again, the correlation function in the integrand of Eq. (38) was expressed as a product of a rotational part and the spin part. However, since it is in this case appropriate to work in the principal frame of the static ZFS, the rotational part becomes proportional to exp(—t/3tb) (if Tfl is the correlation time for reorientation of rank two spherical harmonics, then 3t is the correlation time for rank one spherical... 

The modified Florence program is well-suited for fitting the experimental NMRD profiles for slowly-rotating complexes of gadolinium(HI), an S = 7/2 ion characterized by relatively low ZFS, whose electron spin relaxation can be considered to be in the Redfield limit. An example of fitting an NMRD profile for aqueous protons, using different methods, for a protein adduct of a Gd(HI) chelate capable of accommodating one water molecule in the first coordination sphere, is displayed in Fig. 11. Other examples will be provided in Chapter 3. 

The electron-spin time-correlation functions of Eq. (56) were evaluated numerically by constructing an ensemble of trajectories containing the time dependence of the spin operators and spatial functions, in a manner independent of the validity of the Redfield limit for the rotational modulation of the static ZFS. Before inserting thus obtained electron-spin time-correlation functions into an equation closely related to Eq. (38), Abernathy and Sharp also discussed the effect of distortional/vibrational processes on the electron spin relaxation. They suggested that the electron spin relaxation could be described in terms of simple exponential decay rate constant Ts, expressed as a sum of a rotational and a distortional contribution ... 

Kruk and Kowalewski combined the theory allowing for the radial distribution with their Redfield-limit description of the electron spin relaxation (147). Including the g(r) in the theory led to a more complicated form of the function f(x) of Eq. (69), which becomes dependent on the g(r), as well as on the propagator P(ro, 0/ r, t). The rest of the theory remains unchanged with respect to the presentation in sections VII.A-VII.B. The propagator was computed using the Smoluchowski equation ... 

Tie, the 00sT2e dispersion having already occurred when the (Os v dispersion occurs. Actually, the validity of the SBM theory is assured only within the Redfield limit (see Section IV.A of Chapter 2) (7), i.e., in case the energy of the coupling between the spin and the lattice, E/H (in frequency units), whose modulation is responsible for the spin relaxation, is smaller than the inverse of the correlation time, Xc, for the modulation of the coupling itself, E/H x . This determines for T,e > x (1). 

Investigators should also check, when fitting the relaxivity profile of gadolinium(III) complexes, that electron relaxation is within the Redfield limit (Tie > x ), as otherwise good fits can be still obtained but with parameters that may be wrong by orders of magnitude 10). 

The relaxation equation derived so far for electrons and nuclei share a common assumption usually called the perturbation regime or Redfield limit [54]. The... 

In practice nuclear spin-lattice relaxation is always within the Redfield limit, i.e. the interaction energy with the lattice is always much smaller than rc-1. This is true even with paramagnetic systems, where the nuclear spin-lattice interaction eneigy is often much larger than usual. On the other hand, it is not obvious that electrons are always in the Redfield limit. When electrons are outside the Redfield limit, although nuclear relaxation is in the Redfield limit, it is not easy... 

The absolute value of J, which we have used all over the treatment within the Redfield limit, implies that the theory is the same for ferro- and antiferromagnetic coupling. When J kT, the sign of J is irrelevant for the nuclear shifts, the... 

With the PMK ligand a CoCu derivative has been obtained [7] (Fig. 6.7). From the temperature dependence of the shifts (and magnetic susceptibility measurements in solution), the value of J appears to be positive and much smaller than kT [13]. As expected, the hyperfine shifts are the sum of those of the CuZn and ZnCo systems for each proton (Table 6.2). The NMR lines of the copper domain are now quite sharp, even sharper than those of the cobalt domain (Fig. 6.7). Qualitatively, the data can be accounted for if xs of copper is sizably reduced and approaches that of cobalt (and thus the Redfield limit is reached). 

Redfield limit, and the values for the CH2 protons of his- N,N-diethyldithiocarbamato)iron(iii) iodide, Fe(dtc)2l, a compound for which Te r- When z, rotational reorientation dominates the nuclear relaxation and the Redfield theory can account for the experimental results. When Te Ti values do not increase with Bq as current theory predicts, and non-Redfield relaxation theory (33) has to be employed. By assuming that the spacings of the electron-nuclear spin energy levels are not dominated by Bq but depend on the value of the zero-field splitting parameter, the frequency dependence of the Tj values can be explained. Doddrell et al. (35) have examined the variable temperature and variable field nuclear spin-lattice relaxation times for the protons in Cu(acac)2 and Ru(acac)3. These complexes were chosen since, in the former complex, rotational reorientation appears to be the dominant time-dependent process (36) whereas in the latter complex other time-dependent effects, possibly dynamic Jahn-Teller effects, may be operative. Again current theory will account for the observed Ty values when rotational reorientation dominates the electron and nuclear spin relaxation processes but is inadequate in other situations. More recent studies (37) on the temperature dependence of Ty values of protons of metal acetylacetonate complexes have led to somewhat different conclusions. If rotational reorientation dominates the nuclear and/or electron spin relaxation processes, then a plot of ln( Ty ) against T should be linear with slope Er/R, where r is the activation energy for rotational reorientation. This was found to be the case for Cu, Cr, and Fe complexes with Er 9-2kJ mol" However, for V, Mn, and... 

For Gd(III) the transverse and longitudinal eleetron spin relaxation functions are linear combinations of four decreasing exponentials. As discussed above, for transverse relaxation these exponentials have different weightings and are field dependent. On the other hand, Belorizky and Fries [30] showed that in the Redfield limit of the theory of electronic relaxation, the longitudinal relaxation function has a quasi-monoexponential decay characterized by a unique relaxation rate l/Tje. Neglecting the higher-order terms of the static ZFS, they showed that 1/Tie could be given by a simple analytical expression that is contributed to by static and transient ZFS terms ... 

Rast S, Fries PH, Belorizky E, Borel A, Helm L, Merbach AE. 2001. A general approach to the electronic spin relaxation of Gd(III) complexes in solutions Monte Carlo simulations beyond the Redfield limit. J Chem Phys 115 7554—7563. 

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