Nuclear spin relaxation due to paramagnetic species in solution: Effect of anisotropy in the zero field splitting tensor

The NMR (nuclear magnetic resonance) paramagnetic relaxation enhancement (NMR‐PRE) that is produced by paramagnetic solutes in solution has been investigated theoretically with respect to the influence of zero field splitting (zfs) interactions in the electron spin Hamiltonian, in particular with respect to the effects of anisotropy in the zfs tensor. These effects are a physical consequence of the influence of the zfs on the motion of the electron spin vector S̄. When the zfs energy is large compared to the Zeeman energy (the zfs limit), the precessional motion of S̄ is quantized in the molecule‐fixed coordinate system that diagonalizes the zfs tensor. The uniaxial portion of the zfs tensor influences the NMR‐PRE primarily through its influence on the quantization axes of S̄; the characteristic behavior of the NMR‐PRE under the influence of a uniaxial zfs has been described in detail previously. Anisotropy in the zfs tensor induces oscillatory motion in Sz. This motion has a profound influence on the NMR‐PRE, the major part of which normally arises from low frequency components of the local magnetic field that are associated with Sz, rather than from the rapidly precessing local fields that are associated with the transverse components S±. For this reason, the NMR‐PRE is a sensitive function of zfs anisotropy, which acts to lower the NMR‐PRE below the value that occurs in the uniaxial situation. The magnitude of this effect depends on the ratio (E/D) of the anisotropic and uniaxial zfs parameters, on the reduced dipolar correlation time, and on the location of the nuclear spin in the molecular coordinate frame. A second physical effect of zfs anisotropy on the NMR‐PRE arises from a resonance between the electron spin precessional motion in the transverse plane with the precessional motion that is perpendicular to the transverse plane (the latter due to zfs anisotropy). Resonance of these motions, which occurs spin energy levels crossings, gives rise to low frequency transverse components of S̄ which result in a resonant increase in the NMR‐PRE within a restricted range of E/D ratios.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/71127/2/JCPSA6-98-8-6092-1.pd

Effect of zero field splitting interactions on the paramagnetic relaxation enhancement of nuclear spin relaxation rates in solution

The enhancement of nuclear spin relaxation rate R1m that is produced by paramagnetic metal ions in solution (the NMR‐PRE) has been investigated for electron spin systems with S=1 using recently developed relaxation theory that incorporates both Zeeman and zero field splitting (zfs) interactions of arbitrary magnitude in the electron spin Hamiltonian. The zfs interaction gives rise to important qualitative features which have no analog in the Zeeman‐limit theory. The three principal physical phenomena responsible for these effects are (1) alterations in the geometry of the magnetic dipole–dipole coupling energy due to requantization of the electron spin from laboratory to molecular axes; (2) the crossing or ‘‘pinching’’ of spin energy levels that occurs in the regime of field strengths between the zfs and Zeeman limits; and (3) an effective magnetic field dependence in the reorientational correlation time that results from a change in the appropriate definition of this quantity in the intermediate regime. In the zfs limit and in the intermediate regime, the field dispersion profile depends strongly on the position of the nuclear spin with respect to the molecular coordinate axes. For equatorial positions of the nuclear spin, the principle qualitative feature of the dispersion profile is a strong increase in R1m with increasing field strength coupled, in most cases, with a shallow local R1m maximum; both features are centered near the cross‐over field between the limits. For axial positions, the profile exhibits a feature that is superficially similar to those characteristic of Zeeman‐limit theory, but which is fundamentally different in quantitative properties and in physical origin. As a test of theoretical predictions, the experimental magnetic field profile of the NMR‐PRE of the hexaquo‐Ni(II) cation, an S=1 model system that has previously been studied extensively, has been reinterpreted. It is shown that the major qualitative features of the experimental field profile result specifically from physical effects of the zfs interaction and are closely related to the phenomenon of requantization of the electron spin in the intermediate regime.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/71310/2/JCPSA6-98-2-912-1.pd

Characteristic properties of the nuclear magnetic resonance–paramagnetic relaxation enhancement arising from integer and half‐integer electron spins

The influence of zero field splitting (zfs) interactions on the magnetic field dispersion profile of the nuclear magnetic resonance–paramagnetic relaxation (NMR–PRE) (i.e., the enhancement of nuclear magnetic relaxation rates that is produced by paramagnetic solute species in solution) has been explored systematically for S=1, 3/2, 2, and 5/2 spin systems using recently developed theory. To facilitate comparison of results for different spin values, the theory was expressed in a reduced form with Larmor frequencies in units of ωD (the uniaxial zfs parameter D in rad s−1), and correlation times and spin relaxation times in units of ωD−1. For S=1, the functional form of the profile can be described in terms of five types of qualitative features. Two of these are characteristic of Zeeman‐limit [Solomon, Bloembergen, and Morgan (SBM)] theory and result from the magnetic field dependence of the spin energy level splittings. The remaining three have no analog in Zeeman‐limit theory and arise from a change in the quantization axis of the electron spin precessional motion which, in the zfs limit, lies along molecule‐fixed coordinate axes, and, in the Zeeman limit, lies along the external field direction.The reduced field dispersion profiles for the integer spin systems S=1 and S=2 were found to be very similar to each other, the principal difference being that the midfield positions of the requantization features (types 2, 3, and 4) are shifted for S=2 relative to S=1, the magnitude and sign of the shift depending on the position of the nuclear spin in the molecular coordinate frame. For half‐integer spins, the dispersion profiles exhibit, in addition to the five features characteristic of integer spins, a sixth type of feature, which is centered somewhat to low field of ωSτc=1, where τc is the dipolar correlation time. The type‐6 feature results from field‐dependent level splitting of the mS=±1/2 Kramers doublet. It is present when ωDτc≥1. These theoretical predictions have been examined by means of reinterpretations of the NMR–PRE data for tris‐(acetylacetonato)–metal complexes of V(III) (S=1), Cr(III) (S=3/2), Mo(III) (S=3/2), Mn(III) (S=2), and Fe(III) (S=5/2). As predicted, type‐6 features are absent for the integer spin complexes, for which the T1 field dispersion profiles are nearly field independent.The experimental profiles were successfully simulated quantitatively by the generalized theory, but not by Zeeman‐limit theory. For the half‐integer spin systems, the predicted zfs‐related type‐6 features appear to be present in the profiles, particularly for Mo(acac)3, for which the data deviate significantly from the Zeeman‐limit profile in a manner that is explained by the generalized theory.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70993/2/JCPSA6-98-4-2507-1.pd

Rotational Diffusion and Magnetic Relaxation of 119Sn in Liquid SnCl4 and SnI4

Longitudinal and transverse relaxation times of 119Sn have been measured as a function of temperature in liquid SnCl4 and SnI4. For SnCl4, T1 is a monotonically decreasing function of temperature and is dominated mechanistically by spin‐rotation over the entire liquid range. In SnI4, T1 passes through a maximum near 190°C. Spin‐rotation and scalar coupling to 127I, respectively govern the relaxation above and below this temperature. T2≪T1T2≪T1 in both liquids; scalar coupling, modulated by relaxation of the halogen isotopes, dominates the transverse relaxation. Dipolar coupling does not contribute appreciably to relaxation in either compound. Knowledge of the scalar contributions to T1 and T2 in SnI4 permit calculation of the 127I relaxation time [τ127=1.5(10−7) sec at 150°C] and the angular correlation time [τθ = 3.67(10−12) sec[τθ=3.67(10−12)sec at 150°C]. These values, and the published τ35 for SnCl4, give the tin‐halogen scalar coupling constants: J(119Sn☒35Cl)=470 Hz and J(119Sn☒127I)=940 Hz. Spin‐rotation constants are obtained using Steele's rotational diffusion theory and are used to calculate an absolute shielding scale for 119Sn. From the shielding constants and the relative resonance frequencies of 119Sn and 1H, the 119Sn magnetic moment is calculated to be (−1.04347±0.00036μN). Rotational correlation times in SnCl4 and SnI4 have been compared with theoretical predictions of the J diffusion and damped diffusion models. SnCl4 shows significant deviations from Hubbard's relation although the motion is diffusive according to both theories. Inertial effects are important for reorientation in SnI4(τJ* ∼ 2 at 220°C),SnI4(τJ*∼2at220°C), and damped diffusion appears to describe reorientation more accurately at these temperatures than does J diffusion. The relative roles of spin‐rotation, scalar coupling, and dipolar coupling as relaxation pathways in other tin‐containing systems are considered. It is concluded that the first two interactions usually dominate dipolar coupling for tin and for other heavy metals.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70337/2/JCPSA6-57-12-5321-1.pd

Field dependence of nuclear magnetic relaxation of 119Sn in SnCl4, SnBr4, and SnI4

Longitudinal and transverse relaxation times of 119Sn have been measured as a function of temperature at several field strengths in SnCl4, SnBr4, and SnI4. T2 in all three liquids is field independent and is governed by scalar coupling to the halogen. T1 is strongly field dependent in SnBr4 and SnI4 and exhibits a minimum with increasing temperature due to competing scalar and spin‐rotation interactions. Coupling constants and correlation times previously computed for SnCl4 and SnI4 have been confirmed by measurements at 3.3 kg. Analysis of the data for SnBr4 yield J(119Sn☒81Br)=920 Hz, T2(81Br)=0.748×10−6 sec at 294°K, and τθ(2)τθ(2)=3.1×10−12 sec at 294°K. Molecular reorientation in SnBr4 is highly unusual in that the reduced angular correlation time τθ* reaches a minimum value owing to the onset of dynamically coherent reorientation at a temperature (80°C) that is not far above the melting point (30°C), and τθ* remains in the inertial rotation region over much of the liquid range. Various other indications of significant dissimilarity of microdynamical behavior in SnCl4, SnBr4, and SnI4 are pointed out. Assuming the validity of the J‐diffusion model, we complete the spin‐rotation and magnetic shielding constants for SnBr4, but these results are not consistent with the chemical shifts and shielding constants previously inferred for SnCl4. The inconsistency is believed to be associated with the extremely small reduced frictional constant of SnBr4 and with the possibly inappropriate use of extended diffusion theory to describe reorientation in this liquid.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/71152/2/JCPSA6-60-3-1149-1.pd

Nuclear spin relaxation in paramagnetic solutions. Effects of large zero‐field splitting in the electron spin Hamiltonian

Expressions are derived describing nuclear spin relaxation in paramagnetic salt solutions under conditions where the electron spin Hamiltonian is dominated by a uniaxial quadratic zero‐field splitting (zfs) interaction. In this situation, the electron spin vector is quantized along molecular axes rather than along the external magnetic field. By expressing the time dependence of the electron spin operators, written in the molecular coordinate frame, in the Heisenberg representation and then transforming these expressions to the laboratory coordinate system, simple closed form expressions for the paramagnetic nuclear relaxation increment have been derived. Electron–nuclear dipole–dipole and scalar relaxation mechanisms are considered. The resulting expressions parallel those of Solomon–Bloembergen–Morgan theory, but are valid in the zfs limit rather than the Zeeman limit. Nuclear relaxation rates in the zfs and Zeeman limits exhibit characteristic qualitative differences, some of which have been noted in earlier studies. Of particular note is the fact that the scalar contribution to T−11p is much larger in the zfs than in the Zeeman limit. In most circumstances, T−11p=T−12p in the zfs limit, while in the Zeeman limit, scalar relaxation usually contributes significantly only to T−12p. A vector model of this phenomenon is suggested. The results are valid for arbitrary values of the electron spin quantum number but they assume that electron spin relaxation is in the Redfield limit, i.e., that the correlation times of the coupling between electron spin and the lattice be short on the time scale of electron spin relaxation. This condition is probably satisfied widely when the static zfs is large.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70036/2/JCPSA6-93-10-6921-1.pd

NMR paramagnetic relaxation due to the S = 5/2S=5∕2 complex, Fe(III)-(tetra-p-sulfonatophenyl)porphyrin: Central role of the tetragonal fourth-order zero-field splitting interaction

The metalloporphyrins, Me-TSPP [Me = Cr(III)Me=Cr(III), Mn(III), Mn(II), Fe(III), and TSPP=meso-(tetra-pp-sulfonatophenyl)porphyrin], which possess electron spins S = 3/2S=3∕2, 2, 5/25∕2, and 5/25∕2, respectively, comprise an important series of model systems for mechanistic studies of NMR paramagnetic relaxation enhancement (NMR-PRE). For these S>1/2S>1∕2 spin systems, the NMR-PRE depends critically on the detailed form of the zero-field splitting (zfs) tensor. We report the results of experimental and theoretical studies of the NMR relaxation mechanism associated with Fe(III)-TSPP, a spin 5/25∕2 complex for which the overall zfs is relatively large (D ≈ 10 cm−1)(D≈10cm−1). A comparison of experimental data with spin dynamics simulations shows that the primary determinant of the shape of the magnetic relaxation dispersion profile of the water proton R1R1 is the tetragonal fourth-order component of the zfs tensor. The relaxation mechanism, which has not previously been described, is a consequence of zfs-induced mixing of the spin eigenfunctions of adjacent Kramers doublets. We have also investigated the magnetic-field dependence of electron-spin relaxation for S = 5/2S=5∕2 in the presence of a large zfs, such as occurs in Fe(III)-TSPP. Calculations show that field dependence of this kind is suppressed in the vicinity of the zfs limit, in agreement with observation.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/87861/2/184501_1.pd

Electron‐ and nuclear‐spin relaxation in an integer spin system, tris‐(acetylacetonato)Mn(iii) in solution

Expressions are derived for the intermolecular contribution to the nuclear‐spin relaxation rate in solutions containing dissolved paramagnetic ions with spin S≥1. The calculation assumes that the electron‐spin Hamiltonian is dominated by a large axial zero‐field splitting, and it accounts for effects of Zeeman interactions to first order. The expressions are used to analyze proton‐spin relaxation of the acetone solvent in solutions of tris‐(acetylacetonato)Mn(iii)/ acetone. The main objective was to measure electron‐spin relaxation times of Mn(iii), which in this complex is a high‐spin, d4 ion with integer spin S=2. Spin‐lattice relaxation measurements were conducted over a range of magnetic field strengths (0.28–1.1 T) where the zero‐field splitting is large compared to the Zeeman energy. Electron‐spin relaxation times of Mn(iii) were found to be 8±2 ps, with little dependence on temperature over the range 215–303 K and on magnetic field strength up to 1.1 T. Use of the assumption that Zeeman splittings dominate zero‐field splittings (Solomon–Bloembergen–Morgan theory) resulted in computed electron‐spin relaxation times that are too short by a factor of 3–4.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/71073/2/JCPSA6-92-10-5892-1.pd

Electron spin relaxation due to reorientation of a permanent zero field splitting tensor

Electron spin relaxation of transition metal ions with spin S ≥ 1S⩾1 results primarily from thermal modulation of the zero field splitting (zfs) tensor. This occurs both by distortion of the zfs tensor due to intermolecular collisions and, for complexes with less than cubic symmetry, by reorientational modulation of the permanent zfs tensor. The reorientational mechanism is much less well characterized in previous work than the distortional mechanism although it is an important determinant of nuclear magnetic resonance (NMR) paramagnetic relaxation enhancement phenomena (i.e., the enhancement of NMR relaxation rates produced by paramagnetic ions in solution or NMR-PRE). The classical density matrix theory of spin relaxation does not provide an appropriate description of the reorientational mechanism at low Zeeman field strengths because the zero-order spin wave functions are stochastic functions of time. Using spin dynamics simulation techniques, the time correlation functions of the spin operators have been computed and used to determine decay times for the reorientational relaxation mechanism for S = 1.S=1. In the zfs limit of laboratory field strengths (HZeem≪Hzfs∘),(HZeem≪Hzfs∘), when the zfs tensor is cylindrical, the spin decay is exponential, the spin relaxation time, τS∘ ≈ 0.53τR(1),τS∘≈0.53τR(1), where τR(1)τR(1) is the reorientational correlation time of a molecule-fixed vector. The value of τS∘τS∘ is independent of the magnitude of the cylindrical zfs parameter (D), but it depends strongly on low symmetry zfs terms (the E/DE/D ratio). Other spin dynamics (SD) simulations examined spin decay in the intermediate regime of field strengths where HZeem ≈ Hzfs∘,HZeem≈Hzfs∘, and in the vicinity of the Zeeman limit. The results demonstrate that the reorientational electron spin relaxation mechanism is often significant when Hzfs∘ ≥ HZeem,Hzfs∘⩾HZeem, and that its neglect can lead to serious errors in the interpretation of NMR-PRE data. © 2004 American Institute of Physics.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70866/2/JCPSA6-121-11-5387-1.pd

Nuclear magnetic resonance relaxation enhancements produced by paramagnetic solutes: Effects of rhombicity in the zero field splitting tensor with the S=2 spin system as an example

Effects due to the nonuniaxial part of the zero field splitting (ZFS) tensor on NMR relaxation enhancements produced by paramagnetic species in solution (the NMR PRE) has been studied theoretically and experimentally in the ZFS limit, i.e., in the limit where the ZFS energy is large compared to the Zeeman energy. In the ZFS limit, the precessional motion of the electron spin is quantized with respect to molecule‐fixed coordinate axes. The uniaxial part of the ZFS tensor induces precessional motion in the transverse (x,y) components of the electron spin vector S, and x,y anisotropy in the ZFS tensor (i.e., a nonzero ZFS parameter E) induces precessional motion in the z component of S. The NMR‐PRE phenomenon is particularly sensitive to the motion of Sz and hence also to ZFS anisotropy in the xy plane. Mathematical expressions have been derived which describe the motion of the spin vector evolving under the influence of a general rhombic ZFS Hamiltonian and the influence of this motion on the NMR PRE in the ZFS limit. It is shown that oscillations in Sz occur at the transition frequencies of the S spin system; the frequencies and amplitudes of the precessional components of Sz can be calculated by diagonalizing the general ZFS Hamiltonian. These motions and their consequences with respect to the behavior of the NMR PRE are described in detail for the S=2 spin system. A parametrization of NMR‐PRE data is proposed which gives a clear criterion for the conditions under which rhombic parts of the ZFS tensor significantly affect the relaxation enhancements produced by an S=2 spin system. This criterion is of considerable practical importance for the analysis of NMR‐PRE data, since it defines conditions under which data may be analyzed without the need for independent experimental information concerning the magnitude of the ZFS tensor.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70531/2/JCPSA6-99-1-18-1.pd