Transient Zero-field Splitting Research Articles

An ab initio CASSCF study of zero field splitting fluctuations in the octet ground state of aqueous [Gd(iii)(HPDO3A)(H2O)

In this work, we present ab initio calculations of the zero-field splitting (ZFS) of a gadolinium complex [Gd(iii)(HPDO3A)(H2O)] sampled from an ab initio molecular dynamics (AIMD) simulation. We perform both post-Hartree-Fock (complete active space self-consistent field-CASSCF) and density functional theory (DFT) calculations of the ZFS and compare and contrast the methods with experimental data. Two different density functional approximations (TPSS and LC-BLYP) were investigated. The magnitude of the ZFS from the CASSCF calculations is in good agreement with experiment, whereas the DFT results in varying degrees overestimate the magnitude of the ZFS for both functionals and exhibit a strong functional dependence. It was found in the sampling over the AIMD trajectory that the fluctuations in the transient ZFS tensor derived from DFT are not correlated with those of CASSCF nor does the magnitude of the ZFS from CASSCF and DFT correlate. From the fluctuations in the ZFS tensor, we extract a correlation time of the transient ZFS which is on the sub-picosecond time scale, showing a faster decay than experimental estimates.

Transient versus Static Electron Spin Relaxation in Mn(2+) Complexes Relevant as MRI Contrast Agents.

The zero-field splitting (ZFS) parameters of the [Mn(EDTA)(H2O)](2-)·2H2O and [Mn(MeNO2A)(H2O)]·2H2O systems were estimated by using DFT and ab initio CASSCF/NEVPT2 calculations (EDTA = 2,2',2″,2‴-(ethane-1,2-diylbis(azanetriyl))tetraacetate; MeNO2A = 2,2'-(7-methyl-1,4,7-triazonane-1,4-diyl)diacetate). Subsequent molecular dynamics calculations performed within the atom-centered density matrix propagation (ADMP) approach provided access to the transient and static ZFS parameters, as well as to the correlation time of the transient ZFS. The calculated ZFS parameters present a reasonable agreement with the experimental values obtained from the analysis of (1)H relaxation data. The correlation times calculated for the two systems investigated turned out to be very short (τc ∼ 0.02-0.05 ps), which shows that the transient ZFS is modulated by molecular vibrations. On the contrary, the static ZFS is modulated by the rotation of the complexes in solution, which for the small complexes investigated here is characterized by rotational correlation times of τR ∼ 35-60 ps. As a result, electron spin relaxation in small Mn(2+) complexes is dominated by the static ZFS.

Determination of the Static Zero‐Field Splitting of Gd3+ Complexes in Solution from the Shifts of the Central Magnetic Fields of Their EPR Spectra

In principle, the Redfield theory of EPR spectra applies only to fast-rotating complexes with rather small static zero-field splitting (ZFS) terms. However, at sufficiently high frequencies, typically of 35 GHz and above, it predicts values of the central magnetic fields which are surprisingly accurate well beyond its expected applicability range. This remarkable feature is demonstrated by showing that the Redfield EPR spectrum crosses its baseline at the same point as its "exact" simulated counterpart. It is shown that the shift of the central magnetic field with respect to its limiting value in the absence of ZFS terms is often simply proportional to the square of the magnitude of the static ZFS term divided by the spectrometer frequency. This property is used to determine the magnitude of the static ZFS term independently of its fluctuation dynamics and of the presence of the transient ZFS term.

Joint analysis of ESR lineshapes and 1H NMRD profiles of DOTA-Gd derivatives by means of the slow motion theory.

The "Swedish slow motion theory" [Nilsson and Kowalewski, J. Magn. Reson. 146, 345 (2000)] applied so far to Nuclear Magnetic Relaxation Dispersion (NMRD) profiles for solutions of transition metal ion complexes has been extended to ESR spectral analysis, including in addition g-tensor anisotropy effects. The extended theory has been applied to interpret in a consistent way (within one set of parameters) NMRD profiles and ESR spectra at 95 and 237 GHz for two Gd(III) complexes denoted as P760 and P792 (hydrophilic derivatives of DOTA-Gd, with molecular masses of 5.6 and 6.5 kDa, respectively). The goal is to verify the applicability of the commonly used pseudorotational model of the transient zero field splitting (ZFS). According to this model the transient ZFS is described by a tensor of a constant amplitude, defined in its own principal axes system, which changes its orientation with respect to the laboratory frame according to the isotropic diffusion equation with a characteristic time constant (correlation time) reflecting the time scale of the distortional motion. This unified interpretation of the ESR and NMRD leads to reasonable agreement with the experimental data, indicating that the pseudorotational model indeed captures the essential features of the electron spin dynamics.

A theoretical spin relaxation and molecular dynamics simulation study of the Gd(H2O)93+ complex

A theoretical analysis of the paramagnetically enhanced water proton spin-lattice relaxation of a hydrated Gd(3+) ion is combined with Molecular Dynamics (MD) simulations. The electron-proton dipole-dipole correlation function, C(p)(DD)(tau), as well as the pseudo-rotation (PR) model of the transient zero-field splitting (ZFS) are evaluated with the help of the data from MD simulations. The fast local water motion in the first hydration shell, i.e. the wagging and rocking motions, is found not to change the mono exponential character of the dipole correlation function C(p)(DD)(tau), but is important in the time dependence of the transient ZFS interaction. The dynamics of the transient ZFS interaction is modeled as the water-induced electric field gradient tensor at the site of the metal ion. This approach follows the ideas of the pseudo-rotation model, describing the fluctuating zero-field interaction as a constant amplitude in the principal frame but reorienting according to a rotational diffusion equation of motion. The MD results indicate that the pseudo-rotation model gives a multi-exponential correlation function which oscillates at short times and is described by three exponential terms. The time scale is shorter than previously assumed but contain an intermediate time constant (1-2 ps). The electron spin resonance (ESR) spectral width at half height at frequencies of X-band, Q-band, 75 MHz, 150 MHz and 225 MHz can be reproduced at 320 K without any contributions from 4th or 6th rank ZFS interactions. Consequently, there are two mutually inconsistent dynamic models of the ZFS interaction which can describe the water proton T(1)-NMRD (nuclear magnetic resonance dispersion) profile and the field dependent ESR spectra of the hydrated Gd(III) complex equally well.

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.

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).

Electronic relaxation of paramagnetic metal ions and NMR relaxivity in solution: Critical analysis of various approaches and application to a Gd(III)-based contrast agent

The time correlation functions (TCFs) G(alphaalpha(t)[triple bond](Salpha(t)Salpha(0)) (alpha = x,y,z) of the electronic spin components of a complexed paramagnetic metal ion give information about the time fluctuations of its zero-field splitting (ZFS) Hamiltonian due to the random dynamics of the coordination polyhedron. These TCFs reflect the electronic spin relaxation which plays an essential role in the inner- and outer-sphere paramagnetic relaxation enhancements of the various nuclear spins in solution. When a static ZFS Hamiltonian is allowed by symmetry, its modulation by the random rotational motion of the complex has a great influence on the TCFs. We discuss several attempts to describe this mechanism and show that subtle mathematical pitfalls should be avoided in order to obtain a theoretical framework, within which reliable adjustable parameters can be fitted through the interpretation of nuclear-magnetic relaxation dispersion experimental results. We underline the advantage of the numerical simulation of the TCFs, which avoids the above difficulties and allows one to include the effect of the transient ZFS for all the relative magnitudes of the various terms in the electron-spin Hamiltonian and arbitrary correlation times. This method is applied for various values of the magnetic field taken to be along the z direction. At low field, contrary to previous theoretical expectations, if the transient ZFS has negligible influence, the longitudinal TCF GII(t) [triple bond] G(zz)(t) has a monoexponential decay with an electronic relaxation time T1e different from 1/(2D(r)), D(r) being the rotational diffusion coefficient of the complex. At intermediate and high field, the simulation results show that GII (t) still has a monoexponential decay with a characteristic time T1e, which is surprisingly well approximated by a simple analytical expression derived from the Redfield perturbation approximation of the time-independent Zeeman Hamiltonian, even in the case of a strong ZFS where this approximation is expected to fail. These results are illustrated for spins S = 1, 3/2, and 5/2 in axial and rhombic symmetries. Finally, the simulation method is applied to the reinterpretation of the water-proton relaxivity profile due to P760-Gd(III), an efficient blood pool contrast agent for magnetic-resonance imaging.

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.

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 molecules in the first hydration shell of the Gd3+ ion via 90° pseudorotations [Th. Kowall et al., J. Phys. Chem. 99, 13078 (1995)]. To calculate the longitudinal electronic relaxation function G∥(t) of the Gd3+ ion, its static and transient ZFS parameters in the aqua complex as well as the correlation times of the Brownian rotation and vibrations of this complex are needed. We use the values of these parameters derived from an independent multiple frequency and temperature study of the full electronic paramagnetic resonance spectra of Gd3+ in light water H2O, for magnetic fields where the Redfield limit applies. The predicted NMRD profile is in excellent global agreement with experiment over the whole proton frequency range, especially if the correlation times governing the rotational dynamics of the aqua complex are slightly increased to account for the higher viscosity of D2O with respect to H2O.

Nuclear spin relaxation in solution of paramagnetic complexes with large transient zero-field splitting

A theory for paramagnetic relaxation enhancement (PRE) in systems with S> 1 at low magnetic field is developed for the case when the fluctuations of zero-field splitting (ZFS) in the molecular frame are larger than the averaged ZFS. The validity limits of the new theory are discussed, and its performance is evaluated by comparisons with the general slow-motion approach. The relevance of the new approach for the proton PRE in aqueous solution of Ni(II) is discussed.

Vibrational motions and nuclear spin relaxation in paramagnetic complexes: Hexaaquonickel(II) as an example

The proton paramagnetic relaxation enhancement (PRE) in an aqueous solution of nickel(II) is described in terms of electron spin relaxation caused by damped vibrational motions of Eg and T2g symmetry. The damped vibrations generate a transient zero-field splitting (ZFS), of variable amplitude and variable principal direction in the molecular frame, and are modeled by the Smoluchowski equation. The parameters of the model are obtained from a combination of two approaches: first, quantum-chemical calculations of the ZFS as a function of the geometry of the coordination shell of the nickel(II) ion and, second, molecular-dynamic simulations generating a trajectory of water positions around the metal. The description of the electron spin dynamics is included in the calculations of the PRE in two ways: Using the traditional Solomon–Bloembergen–Morgan approach and also by means of the more general slow-motion theory. The calculated PRE as a function of the magnetic field, free of any adjustable parameters, is compared with the experimental data. The two methods of calculating the PRE agree with each other—and with the experimental data—at high magnetic field. At low field, the models predict very different PRE, and only the general model is in reasonable agreement with the experiments.

Nuclear spin relaxation in paramagnetic systems with zero-field splitting and arbitrary electron spinElectronic Supplementary Information available. See http://www.rsc.org/suppdata/cp/b1/b106659p/

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.

EPR spectroscopy of MRI-related Gd(III) complexes: simultaneous analysis of multiple frequency and temperature spectra, including static and transient crystal field effects.

For the first time, a very general theoretical method is proposed to interpret the full electron paramagnetic resonance (EPR) spectra at multiple temperatures and frequencies in the important case of S-state metal ions complexed in liquid solution. This method is illustrated by a careful analysis of the measured spectra of two Gd3+ (S = 7/2) complexes. It is shown that the electronic relaxation mechanisms at the origin of the EPR line shape arise from the combined effects of the modulation of the static crystal field by the random Brownian rotation of the complex and of the transient zero-field splitting. A detailed study of the static crystal field mechanism shows that, contrarily to the usual global models involving only second-order terms, the fourth and sixth order terms can play a non-negligible role. The obtained parameters are well interpreted in the framework of the physics of the various underlying relaxation processes. A better understanding of these mechanisms is highly valuable since they partly control the efficiency of paramagnetic metal ions in contrast agents for medical magnetic resonance imaging (MRI).

Nuclear spin relaxation in paramagnetic complexes of S=1: Electron spin relaxation effects

Electron spin relaxation for an S=1 system and its field dependence in the presence of static zero-field splitting (ZFS) has been described and incorporated in a model for nuclear spin-lattice relaxation in paramagnetic complexes in solution, proposed earlier by the group in Florence. Slow reorientation is assumed and the electron spin energy level structure (at any orientation of the molecule with respect to the laboratory frame) is described in terms of the Zeeman interaction and of the static ZFS. The electron spin relaxation is assumed to be caused by a transient ZFS modulated by the deformation of the complex described as a distortional (or pseudorotational) motion and the Redfield theory is used to derive the electron spin relaxation matrices. In the description of the electron spin relaxation we neglect any contribution from mechanisms involving modulation by reorientation, such as those of the static ZFS and the less important Zeeman interaction, as we limit ourselves to the slow-rotation limit (i.e., τR≫τS). This in general covers the behavior of proteins and macromolecules. The decomposition (DC) approximation is used, which means that the reorientational motion and electron spin dynamics are assumed to be uncorrelated. This is not a serious problem, due to the slow-rotation condition, since reorientational and distortional motions are time-scale separated. The resulting nuclear magnetic relaxation dispersion (NMRD) profiles obtained using the Florence model are calculated and compared with the calculations of the Swedish approach, which can be considered essentially exact within the given set of assumed interactions and dynamic processes. That theory is not restricted by the Redfield limit and can thus handle electron spin relaxation in the slow-motion regime, which is a consequence of not explicitly defining any electron spin relaxation times. Furthermore, the DC approximation is not invoked, and in addition, the electron spin relaxation is described by reorientationally modulated static ZFS and Zeeman interaction besides the distortionally modulated transient ZFS. The curves computed with the Florence model show a satisfactory agreement with these more accurate calculations of the Swedish approach, in particular for the axially symmetric static ZFS tensor, providing confidence in the adequacy of the electron spin relaxation model under the condition of slow rotation. The comparison is also quite instructive as far as the physical meaning of the electron spin relaxation and of its interplay with the nuclear spin system are concerned.

Slow-motion theory of nuclear spin relaxation in paramagnetic complexes (S=1) of arbitrary symmetry

A generalization of the slow-motion theory of nuclear spin relaxation in paramagnetic systems (S=1) is developed. The new model takes into account the effects of rhombic symmetry in the static zero-field splitting tensor. We also allow the principal axis system of the static zero-field splitting tensor to deviate from the molecule-fixed frame of the dipole–dipole tensor between the nuclear and electron spins. These symmetry-breaking properties have profound effects on the nuclear spin–lattice relaxation rate for some cases. Specifically, the relaxivity is reduced substantially at low magnetic field. Nuclear magnetic relaxation dispersion profiles for a large number of cases are discussed, ranging from slightly asymmetric [low static zero-field splitting (ZFS)] weakly deformable (low transient ZFS) to asymmetric (large static ZFS) highly deformable (large transient ZFS) transition-metal complexes. The dynamical regimes covered for the electron spin range from within the Redfield limit into the slow-motion region. One of the main objectives of this investigation is to provide a standard set of essentially exact calculations using the general slow-motion theory, against which simplified models may be tested.

A low-field paramagnetic nuclear spin relaxation theory

A low-field theory of paramagnetically enhanced proton spin-lattice relaxation is developed resulting in closed analytical expressions. The theory describes paramagnetically enhanced water proton spin-lattice relaxation in complexes of transition metal ions with electron spin quantum number S=1. In the low-field regime the electron spin system is dominated by a static or permanent zero-field splitting interaction which is much larger than the Zeeman interaction. The electron spin is thus quantized in the molecular fixed principal frame of the static ZFS-interaction rather than in the laboratory fixed frame. In the molecular fixed frame the zero-field splitting Hamiltonian is characterized by an axial parameter (D) and a rhombic parameter (E). It is shown how the relative magnitude of D and E strongly influence the magnitude of the paramagnetically enhanced proton spin lattice relaxation rate. In describing electron spin relaxation in the molecular fixed frame, we consider a transient ZFS-interaction and a reorientation modulated Zeeman interaction as the two main relaxation mechanisms. Then, using Redfield theory we derive expressions for the relevant electron spin relaxation rates in terms of spectral densities. In contrast to the Zeeman or high field regime, the electron spin system in the low field limit does not have a permanent magnetic moment. The lack of this magnetic moment implies that electron spin-lattice relaxation processes do not influence the paramagnetically enhanced proton spin-lattice relaxation rates. Instead three spin–spin relaxation rates determine the enhanced proton spin-lattice relaxation profile. We express the electron spin–spin relaxation rates in terms of spectral densities determined by the mean square value of the transient zero-field splitting interaction, its characteristic correlation time, and the reorientational correlation time of rank 1.

Field-Dependent Proton NMR Relaxation in Aqueous Solutions of Ni(II) Ions. A New Interpretation

A new model is presented for nuclear-spin relaxation in paramagnetic transition metal complexes in solution, allowing the electron-spin relaxation to be outside the Redfield limit. The novel feature is that the transient zero-field splitting (ZFS), modulated by distortions of the solvation shell, is allowed to be of rhombic rather than cylindrical symmetry. The model, which assumes that the static ZFS is absent, is applicable to aqueous solutions of transition metal ions. The magnetic-field dependence of the proton spin–lattice relaxation rate enhancement in aqueous solution has been investigated, and calculations are presented for anS= 1 system such as Ni2+(aq), using different degrees of rhombicity of the ZFS and different motional conditions (Redfield limit and slow-motion regime). The new model is also applied to fit the previously reported data for the field dependence of proton relaxation in aqueous solution of Ni(ClO4)2at low pH [J. Kowalewski, T. Larsson, and P.-O. Westlund,J. Magn. Reson.74, 56 (1987)]. The inclusion of rhombicity, motivated by recent theoretical work, provides a model which performs as well as the earlierad hocmodel.

Paramagnetic enhanced proton spin-lattice relaxation in the Ni2+hexa-aquo complex

A reinterpretation of the experimental NMR proton spin-lattice dispersion curve of the Ni2+(H2O)6 complex is presented within a general slow-motion theory. The extended pseudo rotation (PR) model developed allows for cross-correlation effects between the nuclear spin-electron spin dipole-dipole and zero field splitting (ZFS) interaction. It is shown that the decomposition approach, treating the electron spin relaxation and the reorientational dynamic of the dipole-dipole correlation function as independent processes is not generally valid. For the Ni-hexa-aquo complex the transient ZFS interaction and the reorientational correlation time change by about 20 per cent due to the correlation effects. Molecular dynamics (MD) simulation of a divalent ion in water provided the timescale of the dynamics present in the PR and the Smoluchowsky models. The structure and dynamics of the octahedral complex is described. The transient ZFS interaction generated by the low frequency vibration modes n(M ↔ OH2) is characte...