Prior theoretical work has predicted that the NMR paramagnetic relaxation enhancement (NMR-PRE) produced by electron spin S = 1 ions is highly sensitive to orthorhombic terms in the static zero field splitting (zfs) tensor. Zfs orthorhombicity (which implies chemical inequivalence of the three principal directions of the zfs-principal axis system and is described by the zfs E-parameter) is predicted to suppress the NMR-PRE profoundly relative to the reference cylindrical zfs-limit situation. This expectation was tested experimentally by a comparison of the zfs-limit NMR-PRE produced by [Ni(II)(en)3]2+ (en = ethylenediamine), a trigonal complex which lacks zfs-rhombicity, with the zfs-limit NMR-PRE produced by two orthorhombic complexes, [Ni(II)(en)2(H2O)2]2+ and [Ni(II)(en)(H2O)4]2+. As predicted, the zfs-limit NMR-PRE produced by the orthorhombic complexes in the proton resonance of a dioxane probe species in the solvent was strongly suppressed (by factors of approximately 5 and 7, respectively) relative to the comparable measurement on the trigonal complex. The suppression of the NMR-PRE due to the orthorhombic zfs terms is counteracted by an applied Zeeman field, leading to a predicted rise in the NMR-PRE with increasing Zeeman field strength; this rise occurs when the Zeeman energy is comparable to the orthorhombic zfs splitting, 2E. This second prediction of theory was likewise confirmed: the expected rhombicity-induced magnetic field dependence in the NMR-PRE was observed for the orthorhombic complexes but not for the trigonal complex.

Spin dynamic simulation techniques were used to study the influence of static-zero field splitting (zfs) interactions on the NMR paramagnetic relaxation enhancement (PRE) produced by the spin-5/2 complex Mn(II) -TSPP (TSPP = tetrakis(sulfonatophenyl)porphyrinate). The NMR-PRE produced by this complex, recently reported by Bryant ct al. (Inorg. Chem. 1999, 38, 1002), has a magnitude and magnetic field dependence that differ markedly from the predictions of the classical Zeeman-limit theory of Solomon, Bloembergen, and Morgan. We show that this failure results from the influence of the static zfs interaction. Inclusion of a zfs coupling of magnitude D = 0.3 cm(-1) in the electron spin Hamiltonian, in conjunction with a realistic description of the effects of Brownian reorientation on the electron spin motion, produced an accurate fit of the experimental T-1 magnetic field dispersion profile. It is shown that the observed dispersion in the NMR-PRE field dispersion profile results from a change in the axis of spatial quantization from a molecule-fixed to a laboratory-fixed axis that occurs as, with increasing Zeeman-field strength, the zero-order electron spin Hamiltonian changes from the zfs Hamiltonian to the electronic Zeeman Hamiltonian. The results demonstrate the profound importance of the static zfs interaction in analyses of the NMR paramagnetic relaxalion enhancement for S greater than or equal to 1 ions, even in systems where the static zfs is very small (\ D \ < 0.1 cm(-1)). They also illustrate the utility of NMR relaxation data for measuring static zfs couplings.

The NMR paramagnetic relaxation enhancement (NMR-PRE) induced by transition metal ions with electron spin S = 1 is strongly influenced by zero-field splitting (ZFS) interactions when the ZFS Hamiltonian is comparable or greater in magnitude than the electronic Zeeman Hamiltonian (H-ZFS greater than or equal to H-Zeem) In the vicinity of the ZFS limit (H-ZFS much greater than H-Zeem), the spatial quantization of the electron spin motion is oriented along the molecule-fixed principal axes of the ZFS tensor. In this situation, the NMR-PRE, corrected to constant electron-nuclear interspin distance, has been predicted to be a function of the orientation of the electron-nuclear interspin vector with respect to the principal axes of the molecular frame. This prediction was tested experimentally and confirmed for S = 3/2 using the complex, Co-II(acac)(2)(H2O)(2), for which the ZFS-limit axial/equatorial T-1 ratio, rho = 2.7 +/- 0.4, is substantially greater than the Zeeman limit value of unity. A second theoretical prediction has been tested concerning characteristic differences in the effect of ZFS rhombicity on the magnetic field profile of the NMR-PRE produced by integer and half-integer spins. For S = 1, the ZFS-limit NMR-PRE is profoundly depressed due to the effects of the orthorhombic terms of the ZFS tensor, a phenomenon which results physically from the splitting of the \+/-1] non-Kramers doublet by ZFS rhombicity and consequent alterations in the spin eigenfunctions. This depression is reversed by the application of a Zeeman field when the Zeeman energy exceeds the \+/-1] doublet splitting produced by ZFS rhombicity. Thus, the NMR-PRE of orthorhombic S = 1 Ni(II) complexes exhibits a characteristic dispersive feature in which the NMR relaxation efficiency increases several-fold between about 0.2 and 10 T. For half-integer spins, this phenomenon is expected not to be present because the Kramers doublets remain unsplit by ZFS interactions of any magnitude or symmetry. This prediction was tested and confirmed through measurements of the magnetic field dependence of the NMR-PRE produced by the S = 3/2 complex, Co-II(acac)(2)(H2O)(2), the NMR-PRE behavior of which differed qualitatively from that of the analogous S = 1 complex, Ni-II(acac)(2)(H2O)(2), as well as from other previously studied S = 1 model compounds.

Dissolved paramagnetic ions generally provide an efficient mechanism for the relaxation of nuclear spins in solution, a phenomenon called the nuclear magnetic resonance-paramagnetic relaxation enhancement (NMR-PRE). Metal ions with electron spins S⩾1 exhibit rich NMR relaxation phenomena originating in the properties of the zero-field splitting (zfs) interaction, which vanishes for spin-12 ions but which is nonzero for S⩾1 ions in site symmetry lower than cubic. For S⩾1 ions in the vicinity of the zfs-limit, i.e., at magnetic-field strengths low enough that the zfs energy exceeds the Zeeman energy, the NMR-PRE depends strongly on the detailed structure of the electron spin energy levels as well as on the spatial quantization of the spin motion. It is shown theoretically and experimentally that the NMR-PRE produced by integer spins can be influenced strongly by the small intradoublet zero-field splittings, i.e., the splittings between the components of the non-Kramers doublets, which are produced by noncylindrical components of the crystal field potential. These small splittings produce relatively low-frequency oscillations in the dipolar field associated with 〈Sẑ〉 (the spin component along the molecule-fixed ẑ axis). These motions decouple the nuclear spin from the electron spin, thereby depressing, in some cases very strongly, the NMR-PRE. The presence of a relatively small Zeeman field, comparable in magnitude to the intradoublet spacing but small compared to the larger interdoublet zfs splittings, causes a major change in the spin wave functions which has profound effects on the motions of the electron spin. When the Zeeman energy exceeds the small zfs splitting, the oscillatory motion of 〈Sẑ〉 damps out, with the result that the electron spin couples more effectively to the nuclear spin, providing a more efficient NMR relaxation pathway. NMR-PRE data are presented for the S=1 complex Ni(II)(o-pda)2Cl2 (o-pda=ortho-phenylenediamine) which confirm the importance of the splitting of the mS=±1 non-Kramers doublet on the NMR relaxation efficiency. The zfs E-parameter was measured from the NMR data to be |E|=0.26 cm−1. The S=2 spin system, Mn(III)-tetraphenylporphyrin sulfonate, exhibits a related phenomenon which arises from the effects of a small zfs splitting, Δε±2, of the mS=±2 non-Kramers doublet that is caused by a fourfold rotational component of the crystal field potential. The splitting Δε±2 was measured from NMR data to be 0.20 cm−1.

Spin dynamics (SD) methods have been developed to compute NMR paramagnetic relaxation enhancements (NMR-PRE) produced by solutes with electron spin S⩾1 in solution. The spin dynamics algorithms, which are implemented as the computer program SpinDyn.f, are similar in spirit to molecular dynamics calculations in statistical mechanics, except that the spin motion is propagated numerically in time using quantum mechanical equations of motion of the spin operators, rather than Newtonian equations of motion of the molecular degrees of freedom as in MD simulations. SD simulations as implemented in SpinDyn.f provide accurate, flexible, and rapid calculations of NMR-PRE phenomena with few of the assumptions or limitations of previous analytical theories. The program calculates inter- and intramolecular NMR-PRE phenomena for both integer and half-integer spin systems processing under arbitrary Zeeman and zfs Hamiltonians in the presence of Brownian reorientation. Isotropic Brownian reorientation is simulated by means of a finite-step algorithm with adjustable step size. Simulations computed by SpinDyn.f have been used in a systematic study aimed at better understanding the influence of Brownian reorientation on the NMR-PRE of an S=1 ion in a non-Zeeman-limit physical situation. Conditions required for the validity of zfs-limit analytical theory are given. SpinDyn.f has also been used to assess quantitatively the effects of molecular reorientation on a prior analysis of NMR-PRE data for the model S=2 complex ion [tris-(acetylacetonato)manganese(III)] in acetone solution; this system was found to be well described by zfs-limit analytical theory.

Recent theory has predicted that rhombicity in the zero-field splitting (zfs) tensor of transition metal ions with integer spin S greater than or equal to 1 exerts a strong influence on the NMR- paramagnetic relaxation enhancements (NMR-PRE) of resonances of nuclear spins in solution. ZFS rhombicity induces rapid oscillation in the z-component of the electron spin vector, which in the absence of rhombicity, is static with respect to precession or oscillation. Rapid oscillation of S; acts to decouple the nuclear spin magnetic moment from the local field produced by the electron spin, and in this way profoundly depresses the NMR-PRE. The influence of zfs-rhombicity on the solvent H-1 NMR-PRE produced by the complex ion trans-bis(2,4-pentanedione)diaquanickel(II) in dioxane solution has been studied by variable field (0.14-8.5 T) T-1 and T-2 measurements. It is shown that the functional form of the T-1 field dispersion profile can be fit by the mathematical expressions of the Zeeman-limit Solomon-Bloembergen-Morgan theory, although the parameters of such a fit are physically meaningless. Spin dynamics simulation methods which account quantitatively for the effects of zfs interactions lead to a very different physical picture of the spin relaxation process, one in which zfs rhombicity is of central importance in determining the functional form of the field dispersion profile.

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.

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.

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.

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 (xy) components of the electron spin vector S, and xy 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 S(z) 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 S(z) occur at the transition frequencies of the S spin system; the frequencies and amplitudes of the precessional components of S(z) 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.

Expressions for the dipolar nuclear-spin relaxation rates in paramagnetic salt solutions have been derived under conditions where the electronic zero-field splitting (zfs) and Zeeman interactions are of arbitrary magnitude and when electron-spin relaxation is rapid compared to molecular reorientation. The theory is intended to provide continuity between the limiting analytical expressions previously derived for the Zeeman limit [Solomon-Bloembergen-Morgan (SBM) theory] and the zfs limit (R. Sharp, J Chem. Phys. 93, 6921, 1990). The more general solutions parallel the forms of both of these limiting theories in that they are comprised of sums of terms, each term consisting of a mean-square dipolar coupling energy times a spectral density function at one of the transition frequencies of the coupled I-S spin system. Geometric aspects of the problem are exhibited in simplest form in terms of spherical tensors, and the resulting expressions reduce in a straightforward manner to the Zeeman- and zfs-limit equations. As in the limiting theories, the electronspin relaxation time is treated as a parameter of the theory rather than calculated in detail from the time dependence of the electron-spin Hamiltonian. The theory has been applied to the analysis of magnetic field-dependent proton relaxation data of the ligand methyl protons in solutions of tris(acetylacetonato)Mn(III). The agreement with experiment is much superior to that found for SBM theory.

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.

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.

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTField dependence of solvent proton and deuteron NMR relaxation rates of the manganese(II) binding site of chloroplast coupling factor 1Alice E. Haddy and Robert R. SharpCite this: Biochemistry 1989, 28, 9, 3656–3664Publication Date (Print):May 2, 1989Publication History Published online1 May 2002Published inissue 2 May 1989https://pubs.acs.org/doi/10.1021/bi00435a006https://doi.org/10.1021/bi00435a006research-articleACS PublicationsRequest reuse permissionsArticle Views36Altmetric-Citations8LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InRedditEmail Other access optionsGet e-Alertsclose Get e-Alerts