Outer sphere relaxation

PRE effects can also be seen for nuclear spins belonging to the solvent or other molecules present in solution, that essentially do not enter the first coordination sphere of the paramagnetic metal ion. This mechanism is 

Classical studies of the relaxation processes, caused by translational diffusion, have been presented in the early days by Abragam (18), Torrey (136) and Pfeifer (137). Abragam (18) found, by solving the diffusion equation, the following form of the correlation function for the stochastic function Z o under translational diffusion of two spins 1/2  

An important theoretical development for the outer-sphere relaxation was proposed in the 1970s by Hwang and Freed (138). The authors corrected some earlier mistakes in the treatment of the boundary conditions in the diffusion equation and allowed for the role of intermolecular forces, as reflected in the IS radial distribution function, g(r). Ayant et al. (139) proposed, independently, a very similar model incorporating the effects of molecular interactions. The same group has also dealt with the effects of spin eccentricity or translation-rotation coupling (140). 

We defer the discussion of the effects of (r) until Section VII.C and begin with the special case, referred as a force-free diffusion, with a uniform distribution of electron spins outside the distance of closest approach with respect to the nuclear spin. Under the assumption of the reflecting-wall boundary condition at rjs = d, Hwang and Freed found the closed analytical form of the correlation function for translation diffusion (138)  

The Avogadro number, Na, times 1000 times the molar concentration of the metal complex, [Ml, replaces here the spin density N (in units of m ) of Eqs. (67) and (68). 

The diffusional correlation time to depends on the size of both the metal and the ligand-containing moieties, according to their diffusion coefficients, Dm and Di, and on the minimal distance that can be achieved between the ligand and the metal ion, called distance of closest approach, d [8,9]. 

In turn, the diffusion coefficients are defined by assuming that the molecules behave as rigid spheres in a medium of viscosity ry. 

In the diffusion-controlled regime different equations should be derived, taking into account that the interaction eneigy is now modulated by fluctuations in r between d and infinity. In this case the kind of integration to be performed depends on the model assumed for the diffusional behavior of the system. According to one of the most commonly used models for diffusion [11,12], the following equations have been derived when to is the dominant correlation time  

It should be noted that, because the interaction energy is averaged in a different way with respect to the case of shortest zs discussed above, the equations look very different. In particular, the J(io) do not have the usual Lorentzian form (Fig. 4.5). 

Equations are also available for the case of xp and zs having comparable values 

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Fig. 1. Schematic representation of a Gdm complex with one inner sphere water molecule, which is the origin of the inner sphere contribution to proton relaxivity. The complex is surrounded by bulk water, giving rise to the outer sphere relaxation mechanism.

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. 

DR. JACK VRIESENGA (Syracuse University) You pointed out the dangers involved in extracting entropies and enthalpies from NMR data, not only as a result of the cross-correlation between the two, but also their correlation to other NMR parameters. I thought it might be useful for you to comment on the effect of pressure on the other NMR parameters, besides the kinetic control For example, you commented about the role played by the outer-sphere relaxation in the interpretation of NMR relaxation data. How would this be affected by pressure ... 

D. Electron spin dynamics in the equilibrium ensemble Spin-dynamics models Outer-sphere relaxation... 

Bayburt and Sharp 143) formulated a low-field theory (i.e. a theory for the case of ZFS dominating over the electron spin interaction) for the outer-sphere relaxation, treating also the electron spin relaxation in the simplified manner expressed by Eq. (52). That model predicted only a weak dependence of the PRE on the magnitude of the static ZFS and its application to the cases of high static ZFS is problematic. 

As in Eq. (64), the electron spin spectral densities could be evaluated by expanding the electron spin tensor operators in a Liouville space basis set of the static Hamiltonian. The outer-sphere electron spin spectral densities are more complicated to evaluate than their inner-sphere counterparts, since they involve integration over the variable u, in analogy with Eqs. (68) and (69). The main simplifying assumption employed for the electron spin system is that the electron spin relaxation processes can be described by the Redfield theory in the same manner as for the inner-sphere counterpart (95). A comparison between the predictions of the analytical approach presented above, and other models of the outer-sphere relaxation, the Hwang and Freed model (HF) (138), its modification including electron spin... 

Fig. 14. The outer-sphere relaxivity at zero magnetic field as a function of relative diffusion coefficient for S = 1. Reproduced with permission from Kruk, D. Nilsson, T. Kowalewski, J. Mol. Phys. 2001, 99,1435-1445. Copyright 2001 Taylor and Francis Ltd (http //www.tandf.co.uk/journals/tf/00268976.html).

Models for the outer-sphere PRE, allowing for faster rotational motion, have been developed, in analogy with the inner sphere approaches discussed in the Section V.C. The outer-sphere counterpart of the work by Kruk et al. 123) was discussed in the same paper. In the limit of very low magnetic field, the expressions for the outer-sphere PRE for slowly rotating systems 96,144) were found to remain valid for an arbitrary rotational correlation time Tr. New, closed-form expressions were developed for outer-sphere relaxation in the high-field limit. The Redfield description of the electron spin relaxation in terms of spectral densities incorporated into that approach, was valid as long as the conditions A t j 1 and 1 were fulfilled. The validity... 

Abernathy and Sharp (130,145) treated the intermediate regime, when the reorientation of the paramagnetic species is in-between the slow- and fast-rotations limits. They applied the spin-dynamics method, described in Section VI, to the case of outer-sphere relaxation and interpreted NMRD profiles for non-aqueous solvents in the presence of complexes of Ni(II) (S = 1) and Mn(III) (S = 2). 

The outer-sphere relaxation enhancement is another challenging field where further progress is needed. In addition to all the problems met for the inner-sphere cases, one has here to deal with the translational degrees of freedom and with the effects intermolecular forces have upon them. Several important developments were presented during the recent years, as described in Section VII, but much remains to be done. 

Molecular hydration in solution is described not only by the inner-sphere water molecules (first and second coordination spheres, see Section II.A.l) but also by solvent water molecules freely diffusing up to a distance of closest approach to the metal ion, d. The latter molecules are responsible for the so-called outer-sphere relaxation (83,84), which must be added to the paramagnetic enhancement of the solvent relaxation rates due to inner-sphere protons to obtain the total relaxation rate enhancement,... 

The classical equation for 7 sis provided in Section VII.A of Chapter 2. It depends only on the spin quantum number S, on the molar concentration of paramagnetic metal ions, on the distance d, and on a diffusion coefficient D, which is the sum of the diffusion coefficients of both the solvent molecule (Dj) and the paramagnetic complex (Dm), usually much smaller. The outer-sphere relaxivity calculated with this equation at room temperature and in pure water solution, by assuming d equal to 3 A, is shown in Pig. 25. It appears that the dispersions do not have the usual Lorentzian form. 

At variance with the aqua ion, in most manganese(II) proteins and complexes the contact contribution to relaxation is found negligible. This is clearly the case for MnEDTA (Fig. 33), the relaxivity of which indicates the presence of the dipolar contribution only, and one water molecule bound to the complex 93). Actually the profile is very similar to that of GdDTPA (see Chapter 4), and is provided by the sum of inner-sphere and outer-sphere contributions of the same order. The relaxation rate of MnDTPA is accounted for by outer-sphere relaxation only (see Section II.A.7), no water molecules being coordinated to the complex 94). 

In all Mn(II) proteins and in most complexes the contact interaction is found negligible. In fact, the H NMRD profile of MnEDTA, for instance, indicates the presence of the dipolar contribution only, and one water bound to the complex. The relaxation rate of manganese(II) complexes with DTPA (see Fig. 5.56) is instead provided by outer-sphere relaxation only, since no water molecules are bound to the complex (see Section 4.5.2). 

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