Electron-nuclear Dipolar Interaction Research Articles

Rate constants for saturation-recovery EPR and ELDOR of 14N-Spin labels

Saturation-recovery (SR)-EPR can determine electron spin–lattice relaxation rates in liquids over a wide range of effective viscosity, making it especially useful for biophysical and biomedical applications. Here, I develop exact solutions for the SR-EPR and SR-ELDOR rate constants of 14N-nitroxyl spin labels as a function of rotational correlation time and spectrometer operating frequency. Explicit mechanisms for electron spin–lattice relaxation are: rotational modulation of the N-hyperfine and electron-Zeeman anisotropies (specifically including cross terms), spin-rotation interaction, and residual frequency-independent vibrational contributions from Raman processes and local modes. Cross relaxation from mutual electron and nuclear spin flips, and direct nitrogen nuclear spin–lattice relaxation, also must be included. Both the latter are further contributions from rotational modulation of the electron-nuclear dipolar interaction (END). All the conventional liquid-state mechanisms are defined fully by the spin-Hamiltonian parameters; only the vibrational contributions contain fitting parameters. This analysis gives a firm basis for interpreting SR (and inversion recovery) results in terms of additional, less standard mechanisms.

Dynamic Nuclear Polarization with Vanadium(IV) Metal Centers

Summary Dynamic nuclear polarization (DNP) harnesses the large polarization of electron spins to dramatically increase nuclear magnetic resonance (NMR) sensitivity. This study expands the scope of DNP beyond its traditional focus on hyperpolarizing the solvent network using exogenous polarizing agents (PAs). We introduce 1H DNP with endogenous V4+ centers positioned in a set of vanadyl complexes with tunable V4+-1H distances. We traced the polarization transfer from V4+ to 1H spins, specifically differentiating between direct V4+-1Hs polarization transfer and the 1H spin-diffusion-mediated bulk solvent 1H polarization buildup and illuminated the effect of the V4+-1H distance on these processes. These results deepen our understanding of polarization pathways and expand the catalog of PAs to broad-line transition metals. This study establishes crucial first steps toward employing strategically positioned endogenous paramagnetic metal centers for DNP and the conceptual framework of hyperfine DNP spectroscopy that merges both spatial and chemical diagnosis of target nuclear spins.

Coherence transfer and electron T1-, T2-relaxation in nitroxide spin labels

Abragam’s double-commutator spin operator method is used to analyse: 1) electron coherence transfer by intermolecular dipolar interaction between spin-label radicals, and 2) longitudinal and transverse electron spin relaxation by rotational modulation of the Zeeman and nitrogen-hyperfine anisotropies of isolated nitroxide spin labels. Results compatible with earlier treatments by Redfield theory are obtained without specifically evaluating matrix elements. Extension to single-transition operators for isolated nitroxides predicts electron coherence transfer by pseudosecular electron-nuclear dipolar interaction, in the absence of intermolecular dipolar coupling. This explains earlier experimental findings that coherence transfer (specifically dispersion-like distortion of the EPR absorption line shape) does not extrapolate to zero at low concentrations of nitroxide spin labels.

NMR relaxation studies in doped poly-3-methylthiophene

NMR relaxation rates $(1/{T}_{1})$, magnetic susceptibility, and electrical conductivity studies in doped poly-3-methylthiophene are reported in this paper. The magnetic susceptibility data show the contributions from both Pauli and Curie spins, with the size of the Pauli term depending strongly on the doping level. Proton and fluorine NMR relaxation rates have been studied as a function of temperature (3--300 K) and field (for protons at 0.9, 9.0, 16.4, and 23.4 T, and for fluorine at 9.0 T). The temperature dependence of ${T}_{1}$ is classified into three regimes: (a) For $Tl(g{\ensuremath{\mu}}_{B}\mathbf{B}/\mathbf{2}{\mathbf{k}}_{\mathbf{B}})$, the relaxation mechanism follows a modified Korringa relation due to electron-electron interactions and disorder. ${}^{1}\mathrm{H}\ensuremath{-}{T}_{1}$ is due to the electron-nuclear dipolar interaction in addition to the contact term. (b) For the intermediate temperature range $(g{\ensuremath{\mu}}_{B}\mathbf{B}/\mathbf{2}{\mathbf{k}}_{\mathbf{B}})lTl{T}_{\mathrm{BPP}}$ (the temperature where the contribution from the reorientation motion to the ${T}_{1}$ is insignificant) the relaxation mechanism is via spin diffusion to the paramagnetic centers. (c) In the high-temperature regime and at low Larmor frequency the relaxation follows the modified Bloembergen, Purcell, and Pound model. ${T}_{1}$ data analysis has been carried out in light of these models depending upon the temperature and frequency range of study. Fluorine relaxation data have been analyzed and attributed to the $\mathrm{P}{\mathrm{F}}_{6}$ reorientation. The cross relaxation among the ${}^{1}\mathrm{H}$ and ${}^{19}\mathrm{F}$ nuclei has been observed in the entire temperature range suggesting the role of magnetic dipolar interaction modulated by the reorientation of the symmetric molecular subgroups. The data analysis shows that the enhancement in the Korringa ratio is greater in a less conducting sample. Intra- and interchain hopping of charge carriers is found to be a dominant relaxation mechanism at low temperature. Frequency dependence of ${T}_{1}^{\ensuremath{-}1}$ on temperature shows that at low temperature $[Tl(g{\ensuremath{\mu}}_{B}\mathbf{B}/\mathbf{2}{\mathbf{k}}_{\mathbf{B}})]$ the system shows three dimensions and changes to quasi one dimension at high temperature. Moreover, a good correlation between electrical conductivity, magnetic susceptibility, and NMR ${T}_{1}$ data has been observed.

Optimization and prediction of the electron-nuclear dipolar and scalar interaction in 1H and 13C liquid state dynamic nuclear polarization.

During the last 10-15 years, dynamic nuclear polarization (DNP) has evolved as a powerful tool for hyperpolarization of NMR and MRI nuclides. However, it is not as well appreciated that solution-state dynamic nuclear polarization is a powerful approach to study intermolecular interactions in solution. For solutions and fluids, the 1H nuclide is usually dominated by an Overhauser dipolar enhancement and can be significantly increased by decreasing the correlation time (τc) of the substrate/nitroxide interaction by utilizing supercritical fluids (SF CO2). For molecules containing the ubiquitous 13C nuclide, the Overhauser enhancement is usually a profile of both scalar and dipolar interactions. For carbon atoms without an attached hydrogen, a dipolar enhancement usually dominates as we illustrate for sp2 hybridized carbons in the fullerenes, C60 and C70. However, the scalar interaction is dependent on a Fermi contact interaction which does not have the magnetic field dependence inherent in the dipolar interaction. For a comprehensive range of molecular systems we show that molecules that exhibit weakly acidic complexation interaction(s) with nitroxides provide corresponding large scalar enhancements. For the first time, we report that sp hybridized (H-C) alkyne systems, for example, the phenylacetylene-nitroxide system exhibit very large scalar dominated enhancements. Finally, we demonstrate for a wide range of molecular systems that the Fermi contact interaction can be computationally predicted via electron-nuclear hyperfine coupling and correlated with experimental 13C DNP enhancements.

Structural characterization of ion-vapor deposited hydrogenated amorphous carbon coatings by solid state 13C nuclear magnetic resonance

In the present study, unique structural heterogeneity was observed in ion-vapor deposited a-C:H coatings by performing 13C MAS and 1H-13C CPMAS experiments on solid state nuclear magnetic resonance devices. Two distinct types of sp2 C clusters were discovered: one of them denoted as sp2 C′ in content of 3–12 at. % was non-protonated specifically localized in hydrogen-absent regions, while the other dominant one denoted as sp2 C″ was hydrogenated or at least proximate to proton spins. On basis of the notably analogous variation of sp2 C′ content and Raman parameters as function of substrate bias voltage in the whole range of 0.5 kV–3.5 kV, a model of nano-clustering configuration was proposed that the sp2 C′ clusters were embedded between sp2 C″ clusters and amorphous sp3 C matrix as trapped interfaces or boundaries where the sp2 carbon bonds were highly distorted. Continuous increase of bias voltage would promote the nano-clustering and re-ordering of dominant sp2 C″ clusters, thus results in a marked decrease of interspace and a change of the content of sp2 C′ clusters. Further investigation on the 13C magnetization recovery showed typical stretched-exponential approximation due to the prominent presence of paramagnetic centers, and the stretched power α varied within 0.6–0.9 from distinct types of sp2 C clusters. Differently, the magnetization recovery of 1H showed better bi-exponential approximation with long and short T1(H) fluctuated within 40–60 ms and 0.1–0.3 ms approximately in content of 80% ± 5% and 20% ± 5%, respectively, varying with various bias voltages. Meanwhile, the interrupted 13C saturation recovery with an interval of short T1(H) showed that most of quick-relaxing protons were localized in sp2 C″ clusters. Such a short T1(H) was only possibly resulted from a relaxation mechanism associated with electron-nuclear dipolar interaction or spinning diffusion, and its partial distribution in sp2 C″ clusters may imply more complex configurations of unpaired electrons in studied a-C:H coatings.

Hyperfine interactions of narrow-line trityl radical with solvent molecules

The electron nuclear dipolar interactions responsible for some dynamic nuclear polarization (DNP) mechanisms also are responsible for the presence formally in CW EPR spectra of forbidden satellite lines in which both the electron spin and a nuclear spin flip. Such lines arising from 1H nuclei are easily resolved in CW EPR measurements of trityl radicals, a popular family of DNP reagents. The satellite lines overlap some of the hyperfine features from 13C in natural abundance in the trityl radical, but their intensity can be easily determined by simple simulations of the EPR spectra using the hyperfine parameters of the trityl radical. Isotopic substitution of 2H for 1H among the hydrogens of the trityl radical and/or the solvent allows the dipolar interactions from the 1H on the trityl radical and from the solvent to be determined. The intensity of the dipolar interactions, integrated over all the 1H in the system, is characterized by the traditional parameter called reff. For the so-called Finland trityl in methanol, the reff values indicate that collectively the 1H in the unlabeled solvent have a stronger integrated dipolar interaction with the unpaired electron spin of the Finland trityl than do the 1H in the radical and consequently will be a more important DNP route. Although reff has the dimensions of distance, it does not correspond to any simple physical dimension in the trityl radical because the details of the unpaired electron spin distribution and the hydrogen distribution are important in the case of trityls.

Determination of Very Rapid Molecular Rotation by Using the Central Electron Paramagnetic Resonance Line

Picosecond rotational correlation times of perdeuterated tempone (PDT) are found in alkane and aromatic liquids by directly using the spectral width of the central electron paramagnetic resonance line. This is done by mathematically eliminating the nonsecular spectral density from the spectral parameter equations, thereby removing the need to assume a particular form for it. This is preferable to fitting a constant correction factor to the spectral density, because such a factor does not fit well in the low picosecond range. The electron-nuclear spin dipolar interaction between the probe and solvent is shown to be negligible for the very rapid rotation of PDT in these liquids at the temperatures of the study. The rotational correlation times obtained with the proposed method generally agree to within experimental uncertainty with those determined by using the traditional parameters. Using the middle line width offers greater precision and smoother trends. Previous work with the central line width is discussed, and past discrepancies are explained as possibly resulting from residual inhomogeneous broadening. The rotational correlation time almost forms a common curve across all of the solvents when plotted with respect to isothermal compressibility, which shows the high dependence of rotation on liquid free volume.

Effect of glassy modes on electron spin–lattice relaxation in solid ethanol

Electron spin–lattice relaxation (SLR) of TEMPO radical was measured in the crystalline and glassy states of deuterated ethanol in the temperature range 5–80K using X-band electron paramagnetic resonance (EPR). The measured SLR rates are higher in the glassy than in crystalline state and the excess SLR rate in glassy state is much lower than in ethanol. This result suggests that extra modes in glassy state, i.e. glassy modes, produce the excess SLR rate via the electron-nuclear dipolar (END) interaction between the electron spin of radical and the matrix protons or deuterons. Using the soft-potential model and assuming the END interaction between the electron spin and the matrix protons, the contributions to SLR rate of various mechanisms of glassy modes were theoretically analyzed. The evaluations of SLR rates in glassy ethanol indicate two main mechanisms of glassy modes: thermally activated relaxation of double-well systems and phonon-induced relaxation of quasi-harmonic local modes. The SLR rates induced by these mechanisms correlate well with the experimental data.

Electron spin-lattice relaxation in solid ethanol: Effect of nitroxyl radical hydrogen bonding and matrix disorder

The electron spin-lattice relaxation of TEMPO and TEMPONE was measured at temperatures between 5 and 80 K in crystalline and glassy ethanol using X-band electron paramagnetic resonance spectroscopy. The experimental data at the lowest temperatures studied were explained in terms of electron-nuclear dipolar interaction between the paramagnetic center and the localized excitations, whereas at higher temperatures low-frequency vibrational modes from the host matrix and Raman processes should be considered. The strong impact of hydrogen bonding between the dopant molecule and ethanol host on the spin relaxation was observed in ethanol glass whereas in crystalline ethanol both paramagnetic guest molecules behaved similarly.

31P magic angle spinning NMR spectroscopy of paramagnetic rare-earth-substituted Keggin and Wells-Dawson solids.

Paramagnetic rare-earth elements have been examined as NMR structural probes in polyoxoanionic solids, which have a variety of applications as luminescent materials that are usually disordered and therefore intractable by traditional structural methods. Thirteen Keggin and Wells-Dawson polyoxotungstates containing substitutions with lanthanides of different effective magnetic moments have been examined by 31P magic angle spinning NMR spectroscopy. The electron-nuclear dipolar interaction dominating the spinning sideband envelopes is determined by the lanthanide's magnetic moment and was found to be a sensitive probe of the nature of the polyoxoanion, of the positional isomerism, and of the ion stoichiometry. Electron-nuclear dipolar anisotropies computed based on the point-dipole approximation are generally in good agreement with the experimental results. The choice of a specific lanthanide as a structural probe can be tailored to the desired distance range between the phosphorus atoms and the paramagnetic centers to be probed. This approach is expected to be particularly useful in the paramagnetic polyoxoanionic materials lacking long-range order.

Probing Local Environments in Paramagnetic Europium-Substituted Keggin Solids by 31P Magic Angle Spinning NMR Spectroscopy

Paramagnetic Eu-substituted Keggin oxopolytungstates crystallize in different forms, determined by the nature of the counterions. The crystal packing is in turn responsible for the variations in the geometry of paramagnetic Eu sites with respect to the anion core. We probed the paramagnetic environments in a series of Eu-substituted Keggin solids, by 31P magic angle spinning NMR spectroscopy. 31P spinning sideband envelopes are dominated by the electron-nuclear dipolar interaction. For the compounds under investigation, both the magnitude and the asymmetry parameter of the electron-nuclear dipolar coupling tensor are sensitive to the mutual arrangements of paramagnetic Eu sites in the crystal lattice. and also report on the stoichiometry of the anion. The electron-nuclear dipolar coupling tensors were calculated from the crystallographic coordinates and the experimentally determined effective magnetic moments, assuming a point dipole approximation. The computed tensors are in very good agreement with the experimental spectra. Furthermore, the P-Eu distance estimates, accurate to within 0.06-0.12 A, can be obtained directly from the magnitude of the electron-nuclear dipolar coupling. This work demonstrates that 31P MAS NMR spectroscopy is a useful probe for investigating local environments in paramagnetic Keggin solids.

Interactions and mobility of copper(II)–imidazole-containing copolymers

Poly(1-vinylimidazole-co-methyl methacrylate) copolymers (PVM) were obtained from copolymerization of 1-vinylimidazole and methyl methacrylate with 2,2 ′-azobisisobutyronitrile as an initiator. The formation of random copolymers was substantiated by the glass transition temperature ( T g) and the proton spin–lattice relaxation time in the rotating frame ( T H 1ρ). Cu(II)-PVM complexes were prepared by mixing tetrahydrofuran solution of PVM and copper sulfate solution. The formation of coordination bond between PVM and Cu 2+ ions was studied using differential scanning calorimetry, infrared and 13C solid-stated nuclear magnetic resonance spectroscopy. A single composition dependent T g was obtained for the PVM copolymers, and that increased with increasing VI content. The T g value of the Cu(II)-PVM complex was much higher than that of the PVM copolymer with the same composition. The T H 1ρ of the VI units and MMA units in the copolymers and complexes had one value, and that in the complexes was much lower than that in the copolymers. The dramatic decrease in T H 1ρ for the Cu(II)-PVM complexes was due to Cu(II) complexation and electron–nuclear dipolar interactions.

Role of exchange interaction in effecting spin-lattice relaxation: Interpretations of data onCr3+inCu2+xCr2xSn2−2xspinel and dangling bonds in amorphous silicon

The particular role played by the exchange interaction modulated by lattice vibrations (phonons) in effecting spin-lattice relaxation in amorphous materials in conjunction with the spin-orbit coupling, along with the role played by the well-known mechanisms, such as coupling to tunneling level states (TLS) via dipolar and Fermi-contact hyperfine interactions, is investigated. From a review of experimental data, it is found that the relaxation rate as effected by the exchange interaction is two to three orders of magnitude faster than that due to coupling with TLS centers. In particular, it is deduced that both the exchange interaction and coupling to TLS centers via Fermi-contact hyperfine, or electron-nuclear dipolar interaction, lead, in general, to a quadratic temperature dependency of the relaxation rate at low temperatures changing into a linear temperature dependency at intermediate temperatures. However, when the exchange interaction is strong, the relaxation rate becomes independent of temperature at higher temperatures due to cross relaxation of a spin with an exchange-coupled pair of spins, coupled to it via the exchange interaction. This analysis is then applied to interpret the spin-lattice relaxation data on (i) ${\mathrm{Cr}}^{3+}$ ions in the ${\mathrm{Cu}}_{2+x}{\mathrm{Cr}}_{2x}{\mathrm{Sn}}_{2\ensuremath{-}2x}$ spinel samples for $x=0.1$ and 0.2 in the semiconductor range $(x<~0.5),$ and (ii) dangling bonds, with spin $\frac{1}{2}$, in amorphous silicon.

NMR study of the diluted magnetic semiconductor alloysCd1−xMnxSe,Cd1−xCoxSe,andCd1−xFexSe

Diluted magnetic semiconductors ${\mathrm{Cd}}_{1\ensuremath{-}x}{\mathrm{Mn}}_{x}\mathrm{Se}$ for $x=0.01,$ ${\mathrm{Cd}}_{1\ensuremath{-}x}{\mathrm{Fe}}_{x}\mathrm{Se}$ for $x=0.01$ and 0.02, and ${\mathrm{Cd}}_{1\ensuremath{-}x}{\mathrm{Co}}_{x}\mathrm{Se}$ for $x=0.006,$ 0.009, and 0.01 were studied in the temperature range of 180--400 K by ${}^{113}\mathrm{Cd}$ magic-angle-spinning NMR spectroscopy. The NMR spectra of ${\mathrm{Cd}}_{1\ensuremath{-}x}{\mathrm{Fe}}_{x}\mathrm{Se}$ and ${\mathrm{Cd}}_{1\ensuremath{-}x}{\mathrm{Co}}_{x}\mathrm{Se}$ contain a set of resonance lines that are shifted away from the line of the undoped CdSe compound by the transferred hyperfine (THF) interaction between the cadmium nuclei and the paramagnetic ions. The temperature dependence of the THF shifts follows the Curie-Weiss law, and the spin-lattice relaxation times of the shifted lines are significantly shorter than that of the CdSe line. The ${}^{113}\mathrm{Cd}$ lines show anisotropies that are smaller than the values evaluated from the dipolar interaction between the paramagnetic ions $(M)$ and their nearest-neighboring cations $(M\ensuremath{-}\mathrm{S}\mathrm{e}\ensuremath{-}\mathrm{C}\mathrm{d}).$ The detected anisotropies are therefore composed of dipolar and hyperfine contributions from next-nearest-neighboring (NNN) cadmium nuclei $(M\ensuremath{-}\mathrm{S}\mathrm{e}\ensuremath{-}\mathrm{C}\mathrm{d}\ensuremath{-}\mathrm{S}\mathrm{e}\ensuremath{-}\mathrm{C}\mathrm{d}).$ The spin-lattice relaxation times of the spectral lines are determined by the electron-nuclear dipolar interaction with NNN cations. The number of observed lines corresponds to the number of nonequivalent NNN cations around each paramagnetic ion. Using the values of the relaxation times and the amplitudes of the lines, it is possible to correlate each line to a well-defined NNN conformation. The NMR spectra of ${\mathrm{Cd}}_{1\ensuremath{-}x}{\mathrm{Mn}}_{x}\mathrm{Se}$ did not show any fine structure similar to that observed in the Co and Fe alloys.

A Double-Modulation ESR Study of Internal Dynamics in the Glassy Polymer Matrix Detected by a Nitroxide Spin Probe

The double-modulation ESR spin-probe technique was employed to study slow-motional dynamics in amorphous polymer matrices of polybutadiene, polyisobutylene, poly(methyl methacrylate), and paraffin. It was shown that the low-lying excitation levels of the polymer matrix, found in the range from 77 to 190 K below the internal spin-probe transition, affect the effective relaxation rate of a perdeuterated nitroxide probe. The interrelationship between the effective relaxation rates and the local molecular dynamics of side-chain groups attached to the polymer backbone was observed. Possible relaxation mechanisms are discussed within the electron–nuclear dipolar interaction.

EPR and ENDOR study on mixed crystals of an indolinone nitroxide radical and the isoelectronic ketone. II. EPR and ENDOR investigation on radical pairs in mixed crystals

The EPR spectrum due to radical pairs was studied in mixed crystals containing 5% of a nitroxide indolinone radical and 95% of the isoelectronic ketone. Each radical pair gives rise to a pair of lines in the wings of the main spectrum. The first order analysis of the angular dependence of the fine splittings gave the electron-electron dipolar tensors due to three different pairs. The comparison of the experimental tensors with the calculated ones allowed us to identify the radical pairs in the crystal structure. The temperature dependencies of the EPR intensities of two types of radical pairs were analysed and the singlet-triplet energy separations were obtained. The magnetic susceptibility of the pure radical crystal has been determined and compared with the results of the EPR of the mixed crystals. An ENDOR investigation was performed on the radical pairs at T = 5K. The spectra can be explained by taking into account the inter radical electron-nuclear dipolar interaction.

Multiple-quantum electron-electron double resonance

Continuous wave electron-electron double resonance involves the irradiation of one line in an EPR spectrum with an intense “pumping” microwave source and the observation of the effect of that irradiation on another line of the spectrum using a weaker “observing” microwave source. In fluid samples, three ELDOR mechanisms that can be responsible for the transfer of saturation from the pumped transition to the observed transition have been identified. These are Heisenberg exchange between colliding paramagnetic molecules ( 1)) nuclear relaxation induced by rotational modulation of electron nuclear dipolar interaction ( 1)) and, in samples undergoing very slow rotational diffusion, spectral diffusion because of rotational diffusion (2, 3). The ELDOR technique and also relevant theory were introduced in Ref. (1). ELDOR is the subject of a monograph (4) and has been recently reviewed (5). The longitudinal detection of electron paramagnetic resonance involves the irradiation of a single homogeneous line by two closely spaced equally intense microwave frequencies. It was shown by workers at the University of Pisa (6, 7) that under this irradiation multiquantum transitions are induced. The M, component of magnetization oscillates at the difference between the microwave frequencies as well as at multiples of the difference. These workers directly detected these Zeeman oscillations with a pick-up coil oriented along B,,. In this paper, we report the transfer of Zeeman oscillations as induced by two closely spaced microwave frequencies that irradiate one EPR transition to another EPR transition because of liquid-phase Heisenberg exchange. Thus, three CW microwave frequencies are incident on the spin system: two closely spaced frequencies (nominally 10 kHz apart at X band) that serve together to pump a transition and a weaker third frequency used to observe another transition. All frequencies are derived from a single klystron by use of a single sideband mixer (SSM) in an unusual configuration (Fig. 1). The two inputs of the SSM are supplied by the two outputs of the Hewlett-Packard 3326A two-channel synthesizer set to two

Polycrystalline ENDOR patterns from centers with axial EPR spectra. General formulas and simple analytic expressions for deriving geometric information from dipolar couplings

We present a formulation for rapid simulations of polycrystalline ENDOR patterns for systems whose EPR spectra exhibit axial-resolved magnetic interactions (g tensor and “central”-nucleus hyperfine couplings). The ENDOR response of interest is from a nucleus with nonaxial hyperfine couplings and unresolved splittings. The equations simplify when the hyperfine tensor of the nucleus being observed in ENDOR has axial symmetry in its principal axis frame, as would occur for an I = 1 2 nucleus ( 1H or 19F) interacting by a through-space, electron-nuclear dipolar interaction. In such cases the field dependence of the ENDOR frequencies can be described by simple analytic functions, well suited to least-squares determination of metrical parameters that define the nuclear position. Sample calculations are given for a proton interacting with a nitroxide radical.

First-principles study of the conduction-electron-mediated interactions between nuclear spins in copper metal.

Motivated by recent success in observing the ordering of copper nuclei at very low temperatures, we have used first-principles electronic structure calculations to evaluate the conduction-electron-mediated coupling strengths between the nuclear spins. Scalar-relativistic wave functions from a self-consistent linear augmented-plane-wave calculation were used to evaluate the contact, dipolar, and orbital electron-nuclear interactions. Besides the isotropic Ruderman-Kittel coupling between nuclei, the electron-nuclear dipolar and orbital interactions give rise to significant anisotropic coupling terms. The symmetry of the anisotropic coupling is not dipolar in form and the significance of this for analysis of NMR line broadening is pointed out. The coupling strengths are in good agreement with various NMR measurements. Using mean-field theory, the overall coupling predicts an ordering wave vector at (2\ensuremath{\pi}/a)(0.87,0,0) and a helical spin structure; however, this state is only 0.6% lower in energy than the state with ordering vector (2\ensuremath{\pi}/a)(1,0,0), which is the one recently observed in neutron scattering experiments. The numerical calculations are not precise enough to accurately determine the ordering vector because of this extremely small difference in energy for states near the zone boundary along the (1,0,0) direction.