Elie Belorizky

Static zero field splitting effects on the electronic relaxation of paramagnetic metal ion complexes in solution

A general theory of the electronic relaxation of an S state complexed paramagnetic metal ion (Mn2+, Gd3+) in solution is developed. Contrarily to the usual assumption, the electronic relaxation is not only due to the effects of the transient zero field splitting, but is strongly influenced by the static crystal field effect which is modulated by the random Brownian rotation of the complex. The electron paramagnetic resonance measured linewidths of three Gd3+ (S=7/2) complexes at various temperatures and fields [D. H. Powell et al. J. Am. Chem. Soc. 118, 9333 (1996)] are well interpreted in the framework of this model and show the contributions of both mechanisms. The fitted crystal field parameters, the correlation times, and the activation energies are in good agreement with their expected values from the underlying processes. Moreover, our interpretation does not require the addition of any field independent contribution to the linewidth like the spin-rotation mechanism. The longitudinal relaxation func...

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.

A general approach to the electronic spin relaxation of Gd(III) complexes in solutions. Monte Carlo simulations beyond the Redfield limit

The time correlation functions of the electronic spin components of a metal ion without orbital degeneracy in solution are computed. The approach is based on the numerical solution of the time-dependent Schrodinger equation for a stochastic perturbing Hamiltonian which is simulated by a Monte Carlo algorithm using discrete time steps. The perturbing Hamiltonian is quite general, including the superposition of both the static mean crystal field contribution in the molecular frame and the usual transient ligand field term. The Hamiltonian of the static crystal field can involve the terms of all orders, which are invariant under the local group of the average geometry of the complex. In the laboratory frame, the random rotation of the complex is the only source of modulation of this Hamiltonian, whereas an additional Ornstein–Uhlenbeck process is needed to describe the time fluctuations of the Hamiltonian of the transient crystal field. A numerical procedure for computing the electronic paramagnetic resonanc...

Relaxation theory of the electronic spin of a complexed paramagnetic metal ion in solution beyond the Redfield limit

The relaxation of the electronic spin S of a paramagnetic metal ion with fully quenched orbital angular momentum in its ground state is investigated in an external magnetic field through a systematic study of the time correlation functions governing the evolution of the statistical operator (density matrix). Let omega0 be the Larmor angular frequency of S. When the relaxation is induced by a time-fluctuating perturbing Hamiltonian hH1(t) of time correlation tauc, it is demonstrated that after a transient period the standard Redfield approximation is relevant to calculate the evolution of the populations of the spin states if parallelH1 parallel2tauc2/(1+omega0(2)tauc2)<<1 and that this transient period becomes shorter than tauc at sufficiently high field for a zero-field splitting perturbing Hamiltonian. This property, proven analytically and confirmed by numerical simulation, explains the surprising success of several simple expressions of the longitudinal electronic relaxation rate 1/T1e derived from the Redfield approximation well beyond its expected validity range parallelH1 paralleltauc<<1. It has favorable practical consequences on the interpretation of the paramagnetic relaxation enhancement of nuclei used for structural and dynamic studies.

The rotational motion and electronic relaxation of the Gd(III) aqua complex in water revisited through a full proton relaxivity study of a probe solute

Recent advances in the design of fast field cycling (FFC) relaxometers make it now possible to explore the nuclear magnetic relaxation dispersion (NMRD) of semidilute nuclei with short relaxation times. The paramagnetic relaxation rate enhancement of the protons of the tetramethylammonium (CH3)4N+ cation due to the intermolecular magnetic dipolar coupling with the electronic spin S=7/2 of [Gd(D2O)8]3+ in heavy water has been measured between 10 kHz and 800 MHz by combining FFC and standard relaxation techniques. In order to interpret the complete paramagnetic NMRD profile, particularly in the low field region, two previously neglected features are taken into account: (i) The evolution beyond the Redfield limit of the electronic relaxation of the spin S is obtained from accurate Monte Carlo simulations. (ii) The time fluctuation of the static zero field splitting (ZFS) is attributed not only to the usual global Brownian rotational diffusion of the complex, but also to the rearrangement of the water molecul...

Enhancement of the water proton relaxivity by trapping Gd3+ complexes in nanovesicles

We present a theoretical model for calculating the relaxivity of the water protons due to Gd(3+) complexes trapped inside nanovesicles, which are permeable to water. The formalism is applied to the characterization of apoferritin systems [S. Aime et al., Angew. Chem., Int. Ed. 41, 1017 (2002); O. Vasalatiy et al., Contrast Media Mol. Imaging 1, 10 (2006)]. The very high relaxivity due to these systems is attributed to an increase of the local viscosity of the aqueous solution inside the vesicles and to an outer-sphere mechanism which largely dominates the inner-sphere contribution. We discuss how to tailor the dynamic parameters of the trapped complexes in order to optimize the relaxivity. More generally, the potential of relaxivity studies for investigating the local dynamics and residence time of exchangeable molecules in nanovesicles is pointed out.

Nuclear and electronic relaxation in lanthanide solutions: (CH3)4N+/Gd3+ repulsive ion pair in D2O

Abstract The longitudinal relaxation rate and self-diffusion coefficient of the tetramethylammonium protons are investigated at 400 MHz in D 2 O solutions of hydrated Gd 3+ paramagnetic impurities, without and with complexing NO 3 − ions. The results are interpreted using the hypernetted chain approximation of the potential of mean force between the repulsive ions, approximated as charged hard spheres in discrete polar and polarizable water. The standard dipolar relaxation formalism of Solomon is valid for the Gd 3+ lanthanide, i.e. its electron relaxation time is much longer than the translational correlation time of the interionic Brownian diffusion. The coordination effect by NO 3 − is analyzed.

NMR investigation of P(EO)(LiClO4)x

Abstract We present a NMR investigation of poly(ethylene oxide)-lithium perchlorate [P(EO)(LiClO 4 ) x ] for x = 0.125. Measurements of the spin-lattice relaxation times vs. temperature of 1 H and 7 Li nuclei were performed both in the laboratory and rotating frames ( T 1 and T 1ϱ ). The most remarkable feature is the lack of any significant difference between the 1 H and 7 Li relaxation behaviour. The temperature dependence of both the lithium and proton correlation times is shown to be similar to the corresponding behaviour of the conductivity. It is then clearly stated that the microscopic motion of the Li + cations is essentially governed by the segmental motion of the chain.

Simple analytical approximation of the longitudinal electronic relaxation rate of Gd(III) complexes in solutions

More and more sophisticated theoretical models have been developped for a correct description of the relaxation of the electronic spin S = 7/2 of the Gd(III) paramagnetic complexes used as contrast agents in magnetic resonance imaging (MRI). Both the static zero field splitting (ZFS) modulated by the random rotation of the complex and the transient ZFS due to the very fast distortion of this entity must be included in these models. This leads to rather complicated analytical expressions, from which it is difficult to evaluate the respective effects of the physically relevant parameters. However, in the Redfield limit of the theory of electronic spin relaxation, we show that the longitudinal relaxation function G∥(t) has a quasi-monoexponential decay characterized by a unique relaxation rate 1/T1e, which has a simple expression in terms of the applied magnetic field B0, of the static and transient ZFS parameters, and of the rotational and vibrational correlation times. For the typical investigated Gd(III) complexes, this expression is shown to have a very satisfactory accuracy for B0 < 10 T. The various physical parameters as well as the range of validity of the relaxation approximation are discussed in detail.

NMR approach of the electronic properties of the hydrated trivalent rare earth ions in solution

Abstract:The electronic properties of the lanthanide ions are systematically investigated through the NMR of the protons of the tetramethylammonium ion , used as a probe in solution. The effective magnetic moments on the ions are obtained by measuring the paramagnetic shift of the proton resonance line due to the demagnetizing field that is proportional to the rare earth concentration. These results allow to estimate the sizes of the total splitting of the crystal field acting on the ground multiplet of the various lanthanides. The measured intermolecular longitudinal relaxation rates of the protons are compared with the computed values, both of the usual Solomon theory and of the Curie mechanism. The measured values are intermediate between those predicted by the two theories. This allows a rather accurate determination of the lifetimes of the electronic levels of the ions, as they little depend on the details of the description for the relative spatial microdynamics of the / repulsive ion pair.