Electron spin relaxation complexes

Electronic relaxation is a crucial and difficult issue in the analysis of proton relaxivity data. The difficulty resides, on the one hand, in the lack of a theory valid in all real conditions, and, on the other hand, by the technical problems of independent and direct determination of electronic relaxation parameters. At low fields (below 0.1 T), electronic relaxation is fast and dominates the correlation time tc in Eq. (3), however, at high fields its contribution vanishes. The basic theory of electron spin relaxation of Gdm complexes, proposed by Hudson and Lewis, uses a transient ZFS as the main relaxation mechanism (100). For complexes of cubic symmetry Bloembergen and Morgan developed an approximate theory, which led to the equations generally... 

Outer sphere relaxation arises from the dipolar intermolecular interaction between the water proton nuclear spins and the gadolinium electron spin whose fluctuations are governed by random translational motion of the molecules (106). The outer sphere relaxation rate depends on several parameters, such as the closest approach of the solvent water protons and the Gdm complex, their relative diffusion coefficient, and the electron spin relaxation rate (107-109). Freed and others (110-112) developed an analytical expression for the outer sphere longitudinal relaxation rate, (l/Ti)os, for the simplest case of a force-free model. The force-free model is only a rough approximation for the interaction of outer sphere water molecules with Gdm complexes. 

The symbol xso denotes the electron spin relaxation time at zero magnetic field, where Ti = and is another correlation time, associated with distortions of the paramagnetic complex caused by molecular collisions. 

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. 

High-symmetry systems discussed in the previous section are scarce. In systems with lower symmetry and S > 1, we must expect a static ZFS, which can have a profound effect on both the electron spin relaxation and the PRE. The treatment of the PRE in systems with static ZFS requires caution. The reorientational motion of the complex modulates the ZFS which can cause the breach of both the Redfield condition for the electron spin relaxation and the assumption that electron spin relaxation and molecular reorientation are statistically independent (the decomposition approximation). One limit where the decomposition approximation is valid is for slowly rotating systems. 

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 group in Grenoble has used the radial distribution function approach in a series of papers on intermolecular relaxation. We wish to mention in particular some of their papers from the 1990s, where the radial distribution functions were obtained through different approximate methods and a relatively simple description of the electron spin relaxation was applied (150-154). This work has also been reviewed (155,156). In a recent communication from the same group, the improved description of the electron spin relaxation in Gd(III) complexes (120,121) was included in the model and applied for... 

The main difficulty in theoretical predictions of the PRE effects is caused by the description of electron spin relaxation. The transient ZFS interaction is usually considered as present in every complex with S> 1 and its modulation is assumed to provide the dominant mechanism for the electron spin relaxation. The commonly used pseudorotation model 27,85,86), described earlier, has two advantages it captures the essential physics of electron spin relaxation (i.e., the fact that relaxation can be caused by motions faster than the overall rotation of the paramagnetic complex, provided that these motions displace the principal axis of the ZFS), and it leads to relatively straightforward mathematical formulation. The assumption that the... 

The quantum alternative for the description of the vibrational degrees of freedom has been commented by Westlund et al. (85). The comments indicate that, to get a reasonable description of the field-dependent electron spin relaxation caused by the quantum vibrations, one needs to consider the first as well as the second order coupling between the spin and the vibrational modes in the ZFS interaction, and to take into account the lifetime of a vibrational state, Tw, as well as the time constant,T2V, associated with a width of vibrational transitions. A model of nuclear spin relaxation, including the electron spin subsystem coupled to a quantum vibrational bath, has been proposed (7d5). The contributions of the T2V and Tw vibrational relaxation (associated with the linear and the quadratic term in the Taylor expansion of the ZFS tensor, respectively) to the electron spin relaxation was considered. The description of the electron spin dynamics was included in the calculations of the PRE by the SBM approach, as well as in the framework of the general slow-motion theory, with appropriate modifications. The theoretical predictions were compared once again with the experimental PRE values for the Ni(H20)g complex in aqueous solution. This work can be treated as a quantum-mechanical counterpart of the classical approach presented in the paper by Kruk and Kowalewski (161). 

In this case the planar complex is diamagnetic and possesses the usual narrow line, high-resolution diamagnetic spectrum. The tetrahedral complex in Td symmetry would possess a 3T ground state. In approximately tetrahedral nickel(II) complexes the orbital angular momentum is incompletely quenched the result is a very short electron spin relaxation time and an NMR spectrum with relatively narrow, paramagnetic shifted resonances. 

Fig. 1. A Gd(III) complex with one inner sphere water molecule, surrounded by bulk water. Inner sphere proton relaxivity is due to interactions between the Gd electron spin and the water protons on the inner sphere water. Outer sphere relaxivity arises from interactions between the Gd electron spin and bulk water protons. rR stands for the rotational correlation time of the molecule, kex for the water/proton exchange rate and 1/T, 2c for the electron spin relaxation rates of the Gd(III)...

The electronic relaxation rates, as described by Bloembergen, Morgan and McLachlan [12], also depend on the magnetic field. For Gd(III) complexes they are usually interpreted in terms of zero field splitting interactions (ZFS). The electronic relaxation rates can be described by the Eqs. (14-16), often called as the Bloembergen-Morgan theory of paramagnetic electron spin relaxation ... 

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