D. Kruk

NMR RELAXATION IN SOLUTION OF PARAMAGNETIC COMPLEXES: RECENT THEORETICAL PROGRESS FOR S ≥ 1

Publisher Summary This chapter focuses on the first place on the phenomenon of paramagnetic relaxation enhancement and mentions recent theoretical developments in the neighboring fields. The first issue that needs to be clarified is the relation between macroscopic, observable properties of nuclear spins, and their microscopic counterparts. In solutions of transition metal ions or complexes, one can commonly consider a situation where the ligands carrying nuclear spins can reside in two types of environment: in the coordination sphere of the paramagnetic metal ion or in the bulk. If the ligand contains only one type of magnetic nuclei or if interactions between nuclear spins can be disregarded, each of the two sites can be characterized by nuclear spin–lattice and spin–spin relaxation times, T 1 and T 2 , respectively.

Comparison of different methods for calculating the paramagnetic relaxation enhancement of nuclear spins as a function of the magnetic field

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.

Nuclear spin relaxation in paramagnetic systems with zero-field splitting and arbitrary electron spin

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.

Solid State Field-Cycling NMR Relaxometry: Instrumental Improvements and New Applications

The paper reviews recent progress in field cycling (FC) NMR instrumentation and its application to solid state physics. Special emphasis is put on our own work during the last 15years on instrumentation, theory and applications. As far as instrumentation is concerned we report on our development of two types of electronical FC relaxometers, a mechanical FC relaxometer and a combination of FC and one-dimensional microimaging. Progress has been achieved with respect to several parameters such as the accessible field and temperature range as well as the incorporation of sample spinning. Since an appropriate analysis of FC data requires a careful consideration of relaxation theory, we include a theory section discussing the most relevant aspects of relaxation in solids which are related to residual dipolar and quadrupolar interactions. The most important limitations of relaxation theory are also discussed. With improved instrumentation and with the help of relaxation theory we get access to interesting new applications such as ionic motion in solid electrolytes, structure determination in molecular crystals, ultraslow polymer dynamics and rotational resonance phenomena.

Translational and rotational diffusion of glycerol by means of field cycling 1H NMR relaxometry.

Field cycling (FC) (1)H NMR relaxometry has been applied to study translational and rotational dynamics of nondeuterated (-h(8)) and partially deuterated (-h(3) and -h(5)) glycerol in a broad temperature range. We demonstrate that a low-frequency excess intensity observed in the relaxation dispersion stems from intermolecular dipole-dipole interactions mediated by translational dynamics, whereas the main relaxation is attributed to rotational dynamics. A theoretical description of the relaxation processes is formulated accounting for (1)H-(1)H as well as (1)H-(2)H relaxation channels for the partially deuterated systems. While the intermolecular spectral density is derived from the force-free-hard-sphere model (Fick diffusion with appropriate boundary conditions) of translational motion, the intramolecular relaxation contribution is described by a Cole-Davidson spectral density. This ansatz reproduces very well the dispersion profiles obtained from FC (1)H NMR. Moreover, the approach allows extracting the diffusion coefficient D, which is in good agreement with results from gradient (1)H NMR. Thus, (1)H NMR relaxometry has the potential to become an alternative method for measuring the diffusion coefficient in viscous liquids.

Evolution of the dynamic susceptibility in molecular glass formers: results from light scattering, dielectric spectroscopy, and NMR.

Although broadly studied, molecular glass formers are not well investigated above their melting point. Correlation times down to 10(-12) s are easily accessible when studying low-T(g) systems by depolarized light scattering, employing a tandem-Fabry-Perot interferometer and a double monochromator. When combining these techniques with state-of-the-art photon correlation spectroscopy (PCS), broad band susceptibility spectra become accessible which can compete with those of dielectric spectroscopy (DS). Comparing the results with those from DS, optical Kerr effect, and NMR, we describe the evolution of the susceptibilities starting from the boiling point T(b) down to T(g), i.e., from simple liquid to glassy dynamics. Special attention is given to the emergence of the excess wing contribution which is also probed by PCS and which signals a crossover of the spectral evolution. The process is attributed to a small-angle precursor process of the α-relaxation, and the apparent probe dependent stretching of the α-process is explained by a probe dependent contribution of the excess wing. Upon cooling, its emergence is linked to a strong decrease of the strength of the fast dynamics which is taken as reorientational analog of the anomaly of the Debye-Waller factor. Many glass formers show in addition a slow β-process which manifests itself rather universally in NMR, in DS, however, with different amplitudes, but not at all in PCS experiments. Finally, a three-parameter function is discussed interpolating τ(α)(T) from T(b) to T(g) by connecting high- and low-temperature dynamics.

Intermolecular relaxation in glycerol as revealed by field cycling 1H NMR relaxometry dilution experiments

(1)H spin-lattice relaxation rates R(1) = 1/T(1) have been measured for partly deuterated glycerol-h(5) diluted in fully deuterated glycerol-h(0) for progressively lower concentrations of glycerol-h(5). By means of the field cycling (FC) technique relaxation dispersion data, R(1)(ω), have been collected for several temperatures in the frequency range of 10 kHz-20 MHz. In order to disclose the spectral shape of the intra- and intermolecular relaxation, extrapolation of the relaxation data to the zero concentration limit has been performed. The paper confirms that the low frequency excess contribution to the total relaxation rate R(1)(ω) previously reported for several liquids is of intermolecular origin and reflects translational motion, whereas the high-frequency part is attributed to molecular rotation. Thus, intra- and intermolecular relaxation contributions are spectrally separated. The intermolecular relaxation itself contains also a contribution from rotational motion, which is due to non-central positions of the interacting nuclei in the molecule. This eccentricity effect is quantitatively reproduced by treating the intermolecular spectral density as a sum of translational-like (described by the free diffusion model) and rotational-like contributions (described by a Cole-Davidson function). Applying frequency-temperature superposition master curves as well as individual relaxation dispersion data, R(1)(ω), are analyzed. It is demonstrated that, in spite of the rotational influence, the translational diffusion coefficients, D(T), can be extracted from the (1)H relaxation dispersion which gives (1)H NMR relaxometry the potential to become a routine technique determining the diffusion coefficient in liquids.

General treatment of paramagnetic relaxation enhancement associated with translational diffusion

A theory of nuclear spin relaxation in isotropic liquids for nuclear spins interacting with electron spins, residing in other molecules (the outer-sphere relaxation), is presented. The approach, valid outside of the Redfield limit for electron spin relaxation, is an extension of the Swedish slow motion theory [Benetis et al., Mol. Phys. 48, 329 (1983); Nilsson and Kowalewski, J. Magn. Reson. 146, 345 (2000)] for inner-sphere relaxation. It is demonstrated that the outer-sphere relaxation rate can be expressed as an integral of a product of a translational diffusion correlation function and a function analogous to the inner-sphere spectral density. A numerical implementation of the theory is described and applied to a large number of realistic parameter sets for S = 7/2 and S = 1, which may correspond to Gd(III) and Ni(II) systems. It is shown that the outer-sphere contribution is relevant and should be included into the analysis of nuclear magnetic relaxation dispersion relaxation profiles, especially for slow relative translational diffusion and fast molecular tumbling.

Nuclear spin relaxation in paramagnetic systems (S⩾1) under fast rotation conditions

A new theoretical model for nuclear spin relaxation in paramagnetic systems in solution has been developed. Fast rotational motion is included in the model, both as a source of modulation of the static zero-field splitting, which provides a mechanism for electron spin relaxation, and as an origin of the stochastic variation of the electron spin-nuclear spin dipole-dipole interaction leading to nuclear spin relaxation. At the limit of low magnetic field, the model is essentially identical to the earlier formulations from our laboratory, but new closed-form expressions are given for the inner- and outer-sphere relaxation at the high-field limit. Numerical comparisons with a general theory are reported for the inner-sphere case. In addition, some nuclear magnetic relaxation dispersion (NMRD) profiles from the literature are considered for systems where experiments have been done with both low-molecular weight paramagnetic complexes and their adducts with proteins. Previously developed theories are used to interpret data for the slowly rotating protein adducts, and good fits of the fast-rotating counterparts are obtained by further adjustment of one or two additional parameters.

Zero-field splitting in nickel(II) complexes: A comparison of DFT and multi-configurational wavefunction calculations

The zero-field splitting (ZFS) is an important quantity in the electron spin Hamiltonian for S = 1 or higher. We report calculations of the ZFS in some six- and five-coordinated nickel(II) complexes (S = 1), using different levels of theory within the framework of the ORCA program package [F. Neese, Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2, 73 (2012)]. We compare the high-end ab initio calculations (complete active space self-consistent field and n-electron valence state perturbation theory), making use of both the second-order perturbation theory and the quasi-degenerate perturbation approach, with density functional theory (DFT) methods using different functionals. The pattern of results obtained at the ab initio levels is quite consistent and in reasonable agreement with experimental data. The DFT methods used to calculate the ZFS give very strongly functional-dependent results and do not seem to function well for our systems.