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.     

Quantitative interpretation of the very fast electronic relaxation of most Ln3+ ions in dissolved complexes

In a reference frame rigidly bound to the complex, we consider two Hamiltonians possibly at the origin of the very fast electronic relaxation of the paramagnetic lanthanide Ln(3+) ions (Ln = Ce to Nd, Tb to Yb), namely the mean (static) ligand-field Hamiltonian and the transient ligand-field Hamiltonian. In the laboratory frame, the bombardment of the complex by solvent molecules causes its Brownian rotation and its vibration-distorsion dynamics governing the fluctuations of the static and transient terms, respectively. These fluctuations are at the origin of electronic relaxation. The electronic relaxation of a Ln(3+) ion is defined by the decays of the time correlation functions (TCFs) of the longitudinal and transverse components of the total angular momentum J of its ground multiplet. The Brownian rotation of the complex and its vibration-distorsion dynamics are simulated by random walks, which enable us to compute the TCFs from first principles. It is shown that the electronic relaxation is governed mainly by the magnitude of the transient ligand-field, and not by its particular expression. The range of expected values of this ligand-field together with the lower limit of relaxation time enforced by the values of the vibration-distortion correlation time in liquids give rise to effective electronic relaxation times which are in satisfactory overall agreement with the experimental data. In particular, these considerations explain why the electronic relaxation times vary little with the coordinating ligand and are practically independent of the external field magnitude.     

Electron spin relaxation of N@C60 in CS2 in CS2

We examine the temperature dependence of the electron spin relaxation times of the molecules N@C60 and N@C70 (which comprise atomic nitrogen trapped within a carbon cage) in liquid CS2 solution. The results are inconsistent with the fluctuating zero-field splitting (ZFS) mechanism, which is commonly invoked to explain electron spin relaxation for S> or =1 spins in liquid solution, and is the mechanism postulated in the literature for these systems. Instead, we find an Arrhenius temperature dependence for N@C60 , indicating the spin relaxation is driven primarily by an Orbach process. For the asymmetric N@C70 molecule, which has a permanent ZFS, we resolve an additional relaxation mechanism caused by the rapid reorientation of its ZFS. We also report the longest coherence time (T2) ever observed for a molecular electron spin, being 0.25 ms at 170 K.     

Practical route to relative diffusion coefficients and electronic relaxation rates of paramagnetic metal complexes in solution by model-independent outer-sphere NMRD. Potentiality for MRI contrast agents

The relaxation of electronic spins S of paramagnetic species is studied by the field-dependence of the longitudinal, transverse, and longitudinal in the rotating frame relaxation rates R1, R2, and R1rho of nuclear spins I carried by dissolved probe solutes. The method rests on the model-independent low-frequency dispersions of the outer-sphere (OS) paramagnetic relaxation enhancement (PRE) of these rates due to the three-dimensional relative diffusion of the complex with respect to the probe solute. We propose simple analytical formulas to calculate these enhancements in terms of the relative diffusion coefficient D, the longitudinal electronic relaxation time T1e, and the time integral of the time correlation function of the I-S dipolar magnetic interaction. In the domain of vanishing magnetic field, these parameters can be derived from the low-frequency dispersion of R1 thanks to sensitivity improvements of fast field-cycling nuclear relaxometers. At medium field, we present various approaches to obtain these parameters by combining the rates R1, R2, and R1rho. The method is illustrated by a careful study of the proton PREs of deuterated water HOD, methanol CH3OD, and tert-butyl alcohol (CH3)3COD in heavy water in the presence of a recently reported nonacoordinate Gd(III) complex. The exceptionally slow electronic relaxation of the Gd(III) spin in this complex is confirmed and used to test the accuracy of the method through the self-consistency of the low- and medium-field results. The study of molecular diffusion at a few nanometer scale and of the electronic spin relaxation of other complexed metal ions is discussed.     

Stochastic Liouville equation treatment of the electron paramagnetic resonance line shape of an S-state ion in solution

The current approaches used for the analysis of electron paramagnetic resonance spectra of Gd3+ complexes suffer from a number of drawbacks. Even the elaborate model of [Rast et al., J. Chem. Phys. 113, 8724 (2000)] where the electron spin relaxation is explained by the modulation of the zero-field splitting (ZFS), by molecular tumbling (the so called static contribution), and deformations (transient contribution), is only readily applicable within the validity range of the Redfield theory [Advances in Magnetic Resonance, edited by J.-S. Waugh (Academic, New York, 1965), Vol. 1, p. 1], that is, when the ZFS is small compared to the Zeeman energy and the rotational and vibrational modulations are fast compared to the relaxation time. Spin labels (nitroxides and transition metal complexes) have been studied for years in systems that violate these conditions. The theoretical framework commonly used in such studies is the stochastic Liouville equation (SLE). The authors shall show how the physical model of Rast et al. can be cast into the SLE formalism, paying special attention to the specific problems introduced by the [Uhlenbeck and Ornstein, Phys. Rev. 36, 823 (1930)] process used to model the transient ZFS. The resulting equations are very general and valid for arbitrary correlation times, magnetic field strength, electron spin S, or symmetry. The authors demonstrate the equivalence of the SLE approach with the Redfield approximation for two well-known Gd3+ complexes.     

Direct calculation of (1)H(2)O T(1) NMRD profiles and EPR lineshapes for the electron spin quantum numbers S = 1, 3/2, 2, 5/2, 3, 7/2, based on the stochastic Liouville equation combined with Brownian dynamics simulation

Direct calculation of electron spin relaxation and EPR lineshapes, based on Brownian dynamics simulation techniques and the stochastic Liouville equation approach (SLE-L) [Mol. Phys., 2004, 102, 1085-1093], is here generalized to high spin systems with spin quantum number S = 3/2, 2, 5/2, 3 and 7/2. A direct calculation method is demonstrated for electron spin-spin and spin-lattice relaxation, S-, X- and Q-band EPR-lineshapes and paramagnetic enhanced water proton T(1)- NMRD profiles. The main relaxation mechanism for the electron spin system is a stochastic second rank zero field splitting (ZFS). Brownian dynamics simulation techniques are used in describing a fluctuating ZFS interaction which comprises two parts namely the "permanent" part which is modulated by isotropic reorientation diffusion, and the transient part which is modulated by fast local distortion, which is also modelled by the isotropic rotation diffusion model. The SLE-L approach present is applicable both in the perturbation (Redfield) regime as well as outside the perturbation regime, in the so called slow motion regime.     

1H NMR relaxation in glycerol solutions of nitroxide radicals: effects of translational and rotational dynamics

(1)H spin-lattice relaxation rates in glycerol solutions of selected nitroxide radicals at temperatures between 200 K and 400 K were measured at 15 MHz and 25 MHz. The frequency and temperature conditions were chosen in such a way that the relaxation rates go through their maximum values and are affected by neither the electron spin relaxation nor the electron-nitrogen nucleus hyperfine coupling, so that the focus could be put on the mechanisms of motion. By comparison with (1)H spin-lattice relaxation results for pure glycerol, it has been demonstrated that the inter-molecular electron spin-proton spin dipole-dipole interactions are affected not only by relative translational motion of the solvent and solute molecules, but also by their rotational dynamics as the interacting spins are displaced from the molecular centers; the eccentricity effects are usually not taken into account. The (1)H relaxation data have been decomposed into translational and rotational contributions and their relative importance as a function of frequency and temperature discussed in detail. It has been demonstrated that neglecting the rotational effects on the inter-molecular interactions leads to non-realistic conclusions regarding the translational dynamics of the paramagnetic molecules.     

Design of Gd(III)-based magnetic resonance imaging contrast agents: static and transient zero-field splitting contributions to the electronic relaxation and their impact on relaxivity

A multiple-frequency (9.4-325 GHz) and variable-temperature (276-320 K) electron paramagnetic resonance (EPR) study on low molecular weight gadolinium(III) complexes for potential use as magnetic resonance imaging (MRI) contrast agents has been performed. Peak-to-peak linewidths Delta Hpp and central magnetic fields have been analyzed within the Redfield approximation taking into account the static zero-field splitting (ZFS) up to the sixth order and the transient ZFS up to the second order. Longitudinal electronic relaxation is dominated by the static ZFS contribution at low magnetic fields (B < 0.3 T) and by the transient ZFS at high magnetic fields (B > 1.5 T). Whereas the static ZFS clearly depends on the nature of the chelating ligand, the transient ZFS does not. For the relatively fast rotating molecules studied water proton relaxivity is mainly limited by the fast rotation and electronic relaxation has only a marked influence at frequencies below 30 MHz. From our EPR results we can conclude that electronic relaxation will have no influence on the efficiency of Gd(III)-based MRI contrast agents designed for studies at very high magnetic fields (B > 3T).     

Nuclear magnetic resonance proton dipolar order relaxation in thermotropic liquid crystals: a quantum theoretical approach

By means of the Jeener-Broekaert nuclear magnetic resonance pulse sequence, the proton spin system of a liquid crystal can be prepared in quasiequilibrium states of high dipolar order, which relax to thermal equilibrium with the molecular environment with a characteristic time (T1D). Previous studies of the Larmor frequency and temperature dependence of T1D in thermotropic liquid crystals, that included field cycling and conventional high-field experiments, showed that the slow hydrodynamic modes dominate the behavior of T1D, even at high Larmor frequencies. This noticeable predominance of the cooperative fluctuations (known as order fluctuations of the director, OFD) could not be explained by standard models based on the spin-lattice relaxation theory in the limit of high temperature (weak order). This fact points out the necessity of investigating the role of the quantum terms neglected in the usual high temperature theory of dipolar order relaxation. In this work, we present a generalization of the proton dipolar order relaxation theory for highly correlated systems, which considers all the spins belonging to correlated domains as an open quantum system interacting with quantum bath. As starting point, we deduce a formulation of the Markovian master equation of relaxation for the statistical spin operator, valid for all temperatures, which is suitable for introducing a dipolar spin temperature in the quantum regime, without further assumptions about the form of the spin-lattice Hamiltonian. In order to reflect the slow dynamics occurring in correlated systems, we lift the usual short-correlation-time assumption by including the average over the motion of the dipolar Hamiltonian together with the Zeeman Hamiltonian into the time evolution operator. In this way, we calculate the time dependence of the spin operators in the interaction picture in a closed form, valid for high magnetic fields, bringing into play the spin-spin interactions within the microscopic time scale. Then, by adopting the spin-temperature density operator to represent the collective state of the spin system, and removing the traditional hypothesis of high temperature, we deduce an expression for the first order quantum contribution to T1D (-1), in terms of spectral densities, with coefficients in form of spin traces. The properties that distinguish our result from the high-temperature T1D (-1) are as follows. (a) It is exclusively associated to cooperative fluctuations. (b) Because of its quantum character, it relies on both considering the lattice degrees of freedom quantum mechanically and including the spin-spin interactions in the microscopic time scale. With regard to the average dipolar Hamiltonian, only the nonsecular part plays a relevant role. (c) Associated with the structure of the spin operator involved in the quantum contribution, a term arises which is proportional to the number of spins in the correlated molecular domains, showing that the quantum contribution may be of macroscopic size in highly correlated systems. When applied to nematic liquid crystals, the new term exhibits the typical nu(-1/2) Larmor frequency dependence through the spectral density of the OFD, in consistence with the experimental results.     

Time evolution of a single spin inhomogeneously coupled to an interacting spin environment

We study the time evolution of a single spin coupled by exchange interaction to an environment of interacting spin bath modeled by the XY Hamiltonian. By evaluating the spin correlator of the single spin, we observed that the decay rate of the spin oscillations strongly depends on the relative magnitude of the exchange coupling between the single spin and its nearest neighbor J(') and coupling among the spins in the environment J. The decoherence time varies significantly based on the relative coupling magnitudes of J and J('). The decay rate law has a Gaussian profile when the two exchange couplings are of the same order J(') approximately J but converts to exponential and then a power law as we move to the regimes of J(')>J and J(')相似文献   

Numerical analysis of relaxation times of multiple quantum coherences in the system with a large number of spins

We study the decay of multiple quantum (MQ) NMR coherences in systems with the large number of equivalent spins. As being created on the preparation period of the MQ NMR experiment, they decay due to the dipole-dipole interactions (DDI) on the evolution period of this experiment. It is shown that the relaxation time decreases with the increase in MQ coherence order (according to the known results) and in the number of spins. We also consider the modified preparation period of the MQ NMR experiment [G. A. Alvarez and D. Suter, Phys. Rev. Lett. 104, 230403 (2010)] concatenating the short evolution periods under the secular DDI Hamiltonian (the perturbation) with the evolution period under the nonsecular averaged two-spin/two-quantum Hamiltonian. The influence of the perturbation on the decoherence rate is investigated for the systems consisting of 200-600 equivalent spins.     

Surface nuclear spin relaxation of 199Hg

We report the results of a detailed study of surface relaxation of 199Hg nuclear spins in paraffin coated cells. From measurements of the magnetic field and temperature dependence of the spin relaxation rates we determine the correlation time for magnetic fluctuations and the surface adsorption energy. The data indicate that surface relaxation is caused by dipolar coupling to paramagnetic sites on the surface. We also observe changes in the spin relaxation rate caused by ultraviolet radiation resonant with the 254 nm transition in Hg.     

Orbital disproportionation and spin crossover as a pseudo Jahn-Teller effect

It is shown that in systems with electronic half-closed-shell configurations of degenerate orbitals, e(2) and t(3) (which have totally symmetric charge distribution), ground state distortions from high-symmetry geometries may occur due to a strong pseudo Jahn-Teller effect (PJTE) in the excited states, resulting also in a novel phenomenon of PJT-induced spin crossover. There is no JTE neither in the ground state term nor in the excited terms (including degenerate terms) of these configurations but a strong PJT mixing between two excited states [((1)E+(1)A) [cross-filled circle] e and ((2)T(1)+(2)T(2)) [cross-filled circle] e in the e(2) and t(3) cases, respectively] pushes down the lower term to cross the ground state of the undistorted system and to form the global minimum with a distorted geometry. The analysis of the electronic structure of this distorted configuration shows that it is accompanied by orbital disproportionation: instead of proportional population of all degenerate orbitals by one electron each (as in the ground state of the undistorted system that follows Hund's rule), two electrons with opposite spins occupy one orbital, resulting in transformations of the type (e(theta);e(epsilon))-->(e(theta)e(theta)) for e(2) and (t(x);t(y);t(z))-->(t(x);t(x);t(z)) for t(3) systems. Since the two geometry configurations, undistorted and distorted, appertain to different electronic terms that have different spin states, the formation of the global minimum with the distorted configuration is accompanied by a spin crossover. Distinguished from the known spin-crossover phenomenon in some transition metal compounds, the two states with different spin in the PJT-induced spin crossover have also different nuclear configurations, undistorted and distorted, that coexist with a relatively small energy difference. The change of configuration reduces significantly the rate of relaxation between the two states; the relaxation is further reduced by the lower spin-orbital coupling in the light-atom systems as compared with transition metal compounds. This means that there may be systems for which the switch between the two states (in both directions) under perturbations may be observed as a single-molecule phenomenon. Systems with half-closed-shell electronic configurations e(2) and t(3) are available in a variety of molecules from different classes, organic and inorganic; the theory is illustrated here by ab initio calculations for a series of molecular systems, including Si(3), Si(3)C, CuF(3), Na(3), Si(4), Na(4), Na(4) (-), and C(60) (3-), which are in agreement with the experimental data available.     

Applying Unconventional Spectroscopies to the Single-Molecule Magnets,Co(PPh3)2X2 (X=Cl,Br, I): Unveiling Magnetic Transitions and Spin-Phonon Coupling

Large separation of magnetic levels and slow relaxation in metal complexes are desirable properties of single-molecule magnets (SMMs). Spin-phonon coupling (interactions of magnetic levels with phonons) is ubiquitous, leading to magnetic relaxation and loss of memory in SMMs and quantum coherence in qubits. Direct observation of magnetic transitions and spin-phonon coupling in molecules is challenging. We have found that far-IR magnetic spectra (FIRMS) of Co(PPh₃)₂X₂ ( Co-X ; X=Cl, Br, I) reveal rarely observed spin-phonon coupling as avoided crossings between magnetic and u-symmetry phonon transitions. Inelastic neutron scattering (INS) gives phonon spectra. Calculations using VASP and phonopy programs gave phonon symmetries and movies. Magnetic transitions among zero-field split (ZFS) levels of the S=3/2 electronic ground state were probed by INS, high-frequency and -field EPR (HFEPR), FIRMS, and frequency-domain FT terahertz EPR (FD-FT THz-EPR), giving magnetic excitation spectra and determining ZFS parameters (D, E) and g values. Ligand-field theory (LFT) was used to analyze earlier electronic absorption spectra and give calculated ZFS parameters matching those from the experiments. DFT calculations also gave spin densities in Co-X , showing that the larger Co(II) spin density in a molecule, the larger its ZFS magnitude. The current work reveals dynamics of magnetic and phonon excitations in SMMs. Studies of such couplings in the future would help to understand how spin-phonon coupling may lead to magnetic relaxation and develop guidance to control such coupling.     

A ULFC method for d(4)(D(2d)) ions and a study of the spin singlets contributions to zero-field splitting of Cr(2+) ions in zinc sulfide crystals

A simple theoretical method is shown to yield a detailed explanation of numerous EPR parameters for a d4 configuration ion in tetragonal ligand field. Using the unified ligand-field-coupling (ULFC) scheme, the formulas relating the microscopic spin Hamiltonian parameters with the crystal structure parameters are derived. On the basis of the theoretical formulas, the 210 x 210 complete energy matrices including all the spin states are constructed within a strong field representation. By diagonalizing the complete energy matrices, the local lattice structure and Jahn-Teller energy of Cr(2+) ions in ZnS:Cr(2+) system have been investigated. It is found that the theoretical results are in good agreement with the experimental values. Moreover, the contributions of the spin singlets to the zero-field splitting (ZFS) parameters of Cr(2+) ions in ZnS crystals are investigated for the first time. The results indicate that the spin singlets contributions to ZFS parameter b(0)(4) is negligible, but the contributions to ZFS parameters b(0)(4) and b(4)(4) cannot be neglected.     

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.     

Multiple‐Frequency and Variable‐Temperature EPR Study of Gadolinium(III) Complexes with Polyaminocarboxylates: Analysis and Comparison of the Magnetically Dilute Powder and the Frozen‐Solution Spectra

An electron paramagnetic resonance (EPR) study of glasses and magnetically dilute powders of [Gd(DTPA)(H₂O)]^2?, [Gd(DOTA)(H₂O)]^?, and macromolecular gadolinate(1?) complexes P792 was carried out at the X‐ and Q‐bands and at 240 GHz (DTPA=diethylenetriaminepentaacetato; DOTA=1,4,7,10‐tetraazacyclododecane‐1,4,7,10‐tetraacetato). The results show that the zero‐field splitting (ZFS) parameters for these complexes are quite different in a powder as compared to the frozen aqueous solution. In several complexes, an inversion of the sign of the axial component D of the zero field splitting is observed, indicating a significant structural change. In contrary to what was expected, powder samples obtained by lyophilization do not allow a more precise determination of the static ZFS parameters. The results obtained in glasses are more relevant to the problem of electron spin relaxation in aqueous solution than those obtained from powders.     

Spin chemical control of photoinduced electron-transfer processes in ruthenium(II)-trisbipyridine-based supramolecular triads: 2. The effect of oxygen, sulfur, and selenium as heteroatom in the azine donor

Nanosecond time-resolved absorption studies in a magnetic field ranging from 0 to 2.0 T have been performed on a series of covalently linked donor(PXZ)-Ru(bipyridine)3-acceptor(diquat) complexes (D-C2+-A2+). In the PXZ moiety, the heteroatom (X = O (oxygen), T (sulfur), and S (selenium)) is systematically varied to study spin-orbit coupling effects. On the nanosecond time scale, the first detectable photoinduced electron-transfer product after exciting the chromophore C2+ is the charge-separated (CS) state, D+-C2+-A+, where an electron of the PXZ moiety, D, has been transferred to the diquat moiety, A2+. The magnetic-field-dependent kinetic behavior of charge recombination (monoexponential at 0 T progressing to biexponential for all three complexes with increasing field) can be quantitatively modeled by the radical pair relaxation mechanism assuming creation of the CS state with pure triplet spin correlation (3CS). Magnetic-field-independent contributions to the rate constant kr of T+/- --> (T0,S) relaxation are about 4.5 x 10(5) s-1 for DCA-POZ and -PTZ (due to a vibrational mechanism) and 3.5 x 10(6) s-1 for DCA-PSZ (due to spin rotational mechanism). Recombination to the singlet ground state is allowed only from the 1CS spin level; spin-forbidden recombination from 3CS seems negligible even for DCA-PSZ. The field dependence of kr (field-dependent recombination) can be decomposed into the contributions of various relaxation mechanisms. For all compounds, the electron spin dipolar coupling relaxation mechanism dominates the field dependence of tau(slow) at fields up to about 100 mT. Spin relaxation due to the g-tensor anisotropy relaxation mechanism accounts for the field dependence of tau(slow) for DCA-PSZ at high fields. For the underlying stochastic process, a very short correlation time of 2 ps has to be assumed, which is tentatively assigned to a flapping motion of the central, nonplanar ring in PSZ. Finally, it has been confirmed by paramagnetic quenching (here Heisenberg exchange) experiments of the magnetic-field effects with TEMPO that all magnetic-field dependencies observed with the present DCA-PSZ systems are indeed due to the magnetic-field dependence of spin relaxation.