The enhancement of the spin-lattice relaxation rate for nuclear spins in a ligand bound to a paramagnetic metal ion [known as the paramagnetic relaxation enhancement (PRE)] arises primarily through the dipole-dipole (DD) interaction between the nuclear spins and the electron spins. In solution, the DD interaction is modulated mostly by reorientation of the nuclear spin-electron spin axis and by electron spin relaxation. Calculations of the PRE are in general complicated, mainly because the electron spin interacts so strongly with the other degrees of freedom that its relaxation cannot be described by second-order perturbation theory or the Redfield theory. Three approaches to resolve this problem exist in the literature: The so-called slow-motion theory, originating from Swedish groups [Benetis et al., Mol. Phys. 48, 329 (1983); Kowalewski et al., Adv. Inorg. Chem. 57, (2005); Larsson et al., J. Chem. Phys. 101, 1116 (1994); T. Nilsson et al., J. Magn. Reson. 154, 269 (2002)] and two different methods based on simulations of the dynamics of electron spin in time domain, developed in Grenoble [Fries and Belorizky, J. Chem. Phys. 126, 204503 (2007); Rast et al., ibid. 115, 7554 (2001)] and Ann Arbor [Abernathy and Sharp, J. Chem. Phys. 106, 9032 (1997); Schaefle and Sharp, ibid. 121, 5387 (2004); Schaefle and Sharp, J. Magn. Reson. 176, 160 (2005)], respectively. In this paper, we report a numerical comparison of the three methods for a large variety of parameter sets, meant to correspond to large and small complexes of gadolinium(III) and of nickel(II). It is found that the agreement between the Swedish and the Grenoble approaches is very good for practically all parameter sets, while the predictions of the Ann Arbor model are similar in a number of the calculations but deviate significantly in others, reflecting in part differences in the treatment of electron spin relaxation. The origins of the discrepancies are discussed briefly.

Field Cycling Nuclear Magnetic Resonance (FC NMR) relaxation studies are reported for three ionic liquids: 1-ethyl-3- methylimidazolium thiocyanate (EMIM-SCN, 220–258 K), 1-butyl-3-methylimidazolium tetrafluoroborate (BMIM-BF₄, 243–318 K), and 1-butyl-3-methylimidazolium hexafluorophosphate (BMIM-PF₆, 258–323 K). The dispersion of ¹H spin-lattice relaxation rate R₁(ω) is measured in the frequency range of 10 kHz–20 MHz, and the studies are complemented by ¹⁹F spin-lattice relaxation measurements on BMIM-PF₆ in the corresponding frequency range. From the ¹H relaxation results self-diffusion coefficients for the cation in EMIM-SCN, BMIM-BF₄, and BMIM-PF₆ are determined. This is done by performing an analysis considering all relevant intra- and intermolecular relaxation contributions to the ¹H spin-lattice relaxation as well as by benefiting from the universal low-frequency dispersion law characteristic of Fickian diffusion which yields, at low frequencies, a linear dependence of R₁ on square root of frequency. From the ¹⁹F relaxation both anion and cation diffusion coefficients are determined for BMIM-PF₆. The diffusion coefficients obtained from FC NMR relaxometry are in good agreement with results reported from pulsed- field-gradient NMR. This shows that NMR relaxometry can be considered as an alternative route of determining diffusion coefficients of both cations and anions in ionic liquids.

A complete theory for paramagnetic relaxation enhancement (PRE) and its dependence on the magnetic field is developed for systems with electron spin S = 1, 3/2, 2, 5/2, 3 and 7/2, characterised by the presence of zero-field splitting (ZFS). The electron spin interacts through dipole–dipole coupling with the nuclear spin residing in the paramagnetic complex (the inner-sphere case) as well as outside of it (the outer-sphere case). The earlier theory for S = 1 and inner-sphere interaction only is included as a special case of the present, more general approach. The theory assumes a slow reorientation of the paramagnetic complex and a lack of correlation between the rotation and translation of the complex and the electron spin dynamics. The electron spin energy level structure is determined by a combination of the Zeeman interaction and the static ZFS, and depends thus on the orientation of the complex in the magnetic field. The electron spin relaxation is described by a Redfield formulation, using the pseudorotation model for the modulation of the transient zero-field splitting. Illustrative calculations are presented, showing that the typical field dependences of the inner- and outer-sphere relaxation enhancements are in general different. The static ZFS is demonstrated to influence the magnetic field dependences by affecting the frequencies occurring in the expressions for spectral densities. The model is applied to interpret the PRE of a slowly-rotating gadolinium(III) complex, a potential magnetic resonance imaging (MRI) contrast agent.