Theory for cross effect dynamic nuclear polarization under magic-angle spinning in solid state nuclear magnetic resonance: The importance of level crossings

We present theoretical calculations of dynamic nuclear polarization (DNP) due to the cross effect in nuclear magnetic resonance under magic-angle spinning (MAS). Using a three-spin model (two electrons and one nucleus), cross effect DNP with MAS for electron spins with a large g-anisotropy can be seen as a series of spin transitions at avoided crossings of the energy levels, with varying degrees of adiabaticity. If the electron spin-lattice relaxation time T(1e) is large relative to the MAS rotation period, the cross effect can happen as two separate events: (i) partial saturation of one electron spin by the applied microwaves as one electron spin resonance (ESR) frequency crosses the microwave frequency and (ii) flip of all three spins, when the difference of the two ESR frequencies crosses the nuclear frequency, which transfers polarization to the nuclear spin if the two electron spins have different polarizations. In addition, adiabatic level crossings at which the two ESR frequencies become equal serve to maintain non-uniform saturation across the ESR line. We present analytical results based on the Landau-Zener theory of adiabatic transitions, as well as numerical quantum mechanical calculations for the evolution of the time-dependent three-spin system. These calculations provide insight into the dependence of cross effect DNP on various experimental parameters, including MAS frequency, microwave field strength, spin relaxation rates, hyperfine and electron-electron dipole coupling strengths, and the nature of the biradical dopants.     

A local spin study for magnetic model complex HHeH

Chemists, physicists and material scientists havegiven great attention to the theoretical studies of mag-netic systems owing to their significance in the fun-damental research areas as well as their potential ap-plications in the field of high technologie…     

Charge-induced spin alignment in diradical donor molecules: numerical calculations of correlated many-electron-spin systems

The mechanism of charge-induced high spin is studied in pi-conjugated molecules by means of a model-Hamiltonian approach. Intersite Coulomb interactions are taken into account in a pi-conjugated moiety, which is coupled with two localized spins through exchange interactions. We clarify spin alignment in neutral and oxidized states by exact numerical calculations including all the correlation effects. In thianthrene-based molecules, one-electron oxidation induces strong ferromagnetic correlation between the localized spins irrespective of the spin alignment in the neutral state. The localized spins are coupled to the delocalized hole spin ferromagnetically, leading to a high-spin state in the oxidized molecule. Our calculations on structural dependence and effective exchange interaction are consistent with the recent experiment of thianthrene bis(nitronyl nitroxide). By comparing the thianthrene-based molecule with the anthracene-based one, we clarify the role of superexchange interactions via the sulfur atoms.     

Stockholder projector analysis: a Hilbert-space partitioning of the molecular one-electron density matrix with orthogonal projectors

A previously introduced partitioning of the molecular one-electron density matrix over atoms and bonds [D. Vanfleteren et al., J. Chem. Phys. 133, 231103 (2010)] is investigated in detail. Orthogonal projection operators are used to define atomic subspaces, as in Natural Population Analysis. The orthogonal projection operators are constructed with a recursive scheme. These operators are chemically relevant and obey a stockholder principle, familiar from the Hirshfeld-I partitioning of the electron density. The stockholder principle is extended to density matrices, where the orthogonal projectors are considered to be atomic fractions of the summed contributions. All calculations are performed as matrix manipulations in one-electron Hilbert space. Mathematical proofs and numerical evidence concerning this recursive scheme are provided in the present paper. The advantages associated with the use of these stockholder projection operators are examined with respect to covalent bond orders, bond polarization, and transferability.     

Spin exchange interactions and magnetic structures of extended magnetic solids with localized spins: theoretical descriptions on formal, quantitative and qualitative levels

Low-energy excitation energies of a magnetic solid with localized spins are probed by magnetic susceptibility, neutron scattering and Raman scattering measurements, and are analyzed using a spin Hamiltonian with a set of spin exchange parameters. The nature and values of the spin exchange parameters deduced from this analysis depend on what spin exchange paths one includes in the spin Hamiltonian. In this article, we review how spin exchange interactions of magnetic solids with localized spins are described on formal, quantitative and qualitative theoretical levels, investigate antisymmetric and anisotropic interactions for general spin dimers, and discuss the spin exchange interactions and magnetic structures of various extended magnetic solids on the basis of spin dimer analysis. Strongly interacting spin exchange paths of a magnetic solid are determined by the overlap between its magnetic orbitals, so that the strongly interacting spin unit of a magnetic solid does not necessarily have the same geometrical feature as does the arrangement of its magnetic ions or spin-carrying molecules. Therefore, in interpreting results of magnetic susceptibility, inelastic neutron scattering or Raman scattering measurements, it is essential to employ a set of spin exchange parameters chosen on the basis of proper electronic structure considerations. Spin dimer analyses based on extended Hückel tight binding calculations provide a reliable and expedient means to study the relative strengths of superexchange and super-superexchange spin exchange interactions.     

Ab initio modeling of CW-ESR spectra of the double spin labeled peptide Fmoc-(Aib-Aib-TOAC)2-Aib-OMe in acetonitrile

In this work we address the interpretation, via an ab initio integrated computational approach, of the CW-ESR spectra of the double spin labeled, 310-helical, peptide Fmoc-(Aib-Aib-TOAC)2-Aib-OMe dissolved in acetonitrile. Our approach is based on the determination of geometric and local magnetic parameters of the heptapeptide by quantum mechanical density functional calculations taking into account solvent and, when needed, vibrational averaging contributions. The system is then described by a stochastic Liouville equation for the two electron spins interacting with each other and with two 14N nuclear spins, in the presence of diffusive rotational dynamics. Parametrization of the diffusion rotational tensor is provided by a hydrodynamic model. CW-ESR spectra are simulated with minimal resorting to fitting procedures, proving that the combination of sensitive ESR spectroscopy and sophisticated modeling can be highly helpful in providing 3D structural and dynamic information on molecular systems.     

Electron spin relaxation of N@C60 in CS2 in CS2

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

Local Spin Analysis and Chemical Bonding

The electronic structure of main‐group diatomic molecules is discussed in the light of local spin analysis. A deep investigation into the origin of local spins and their coupling is presented. It is shown that the presence of significant local spins in bonded molecules flags deviations from the classical bonding prototypes. For the notorious example of the C₂ molecule, the local spin analysis indicates that its ground state has all ingredients to be categorized as a diradical.     

Antiferromagnetic interactions in the quarter-filled organic conductor (EDO-TTF)2PF6

The ground state electronic structure of the high-temperature (HT) and the low-temperature (LT) phases of (EDO-TTF)(2)PF(6) is investigated using the embedded cluster approach in combination with the density functional method designed to describe the strong non-dynamic electron correlation. It is found that, in the HT phase, the unpaired electron spins located on pairs of neighbouring EDO-TTF molecules are antiferromagnetically coupled along the stacking direction with the Heisenberg exchange integral J = -655 cm(-1). In the LT phase, the unpaired spins located on the cationic EDO-TTF molecules are coupled antiferromagnetically with J values strongly alternating along the stacking axis of the crystal thus rendering it diamagnetic. The parameters of the extended Hubbard model are evaluated and the conductance properties of the two phases are estimated using these parameters. It is suggested to investigate the charge and spin excitations in the two phases of (EDO-TTF)(2)PF(6) with the use of angle-resolved photoemission spectroscopy.     

Multireference perturbation theory with optimized partitioning. II. Applications to molecular systems

The second‐order multireference perturbation theory using an optimized partitioning, denoted as MROPT(2), is applied to calculations of various molecular properties—excitation energies, spectroscopic parameters, and potential energy curves—for five molecules: ethylene, butadiene, benzene, N₂, and O₂. The calculated results are compared with those obtained with second‐ and third‐order multireference perturbation theory using the traditional partitioning techniques. We also give results from computations using the multireference configuration interaction (MRCI) method. The presented results show very close resemblance between the new method and MRCI with renormalized Davidson correction. The accuracy of the new method is good and is comparable to that of second‐order multireference perturbation theory using Møller‐Plesset partitioning. © 2003 Wiley Periodicals, Inc. J Comput Chem 24: 1390–1400, 2003     

Imaging mesoscopic nuclear spin noise with a diamond magnetometer

Magnetic resonance imaging can characterize and discriminate among tissues using their diverse physical and biochemical properties. Unfortunately, submicrometer screening of biological specimens is presently not possible, mainly due to lack of detection sensitivity. Here we analyze the use of a nitrogen-vacancy center in diamond as a magnetic sensor for nanoscale nuclear spin imaging and spectroscopy. We examine the ability of such a sensor to probe the fluctuations of the "classical" dipolar field due to a large number of neighboring nuclear spins in a densely protonated sample. We identify detection protocols that appropriately take into account the quantum character of the sensor and find a signal-to-noise ratio compatible with realistic experimental parameters. Through various example calculations we illustrate different kinds of image contrast. In particular, we show how to exploit the comparatively long nuclear spin correlation times to reconstruct a local, high-resolution sample spectrum.     

Dynamic nuclear polarization assisted spin diffusion for the solid effect case

The dynamic nuclear polarization (DNP) process in solids depends on the magnitudes of hyperfine interactions between unpaired electrons and their neighboring (core) nuclei, and on the dipole-dipole interactions between all nuclei in the sample. The polarization enhancement of the bulk nuclei has been typically described in terms of a hyperfine-assisted polarization of a core nucleus by microwave irradiation followed by a dipolar-assisted spin diffusion process in the core-bulk nuclear system. This work presents a theoretical approach for the study of this combined process using a density matrix formalism. In particular, solid effect DNP on a single electron coupled to a nuclear spin system is considered, taking into account the interactions between the spins as well as the main relaxation mechanisms introduced via the electron, nuclear, and cross-relaxation rates. The basic principles of the DNP-assisted spin diffusion mechanism, polarizing the bulk nuclei, are presented, and it is shown that the polarization of the core nuclei and the spin diffusion process should not be treated separately. To emphasize this observation the coherent mechanism driving the pure spin diffusion process is also discussed. In order to demonstrate the effects of the interactions and relaxation mechanisms on the enhancement of the nuclear polarization, model systems of up to ten spins are considered and polarization buildup curves are simulated. A linear chain of spins consisting of a single electron coupled to a core nucleus, which in turn is dipolar coupled to a chain of bulk nuclei, is considered. The interaction and relaxation parameters of this model system were chosen in a way to enable a critical analysis of the polarization enhancement of all nuclei, and are not far from the values of (13)C nuclei in frozen (glassy) organic solutions containing radicals, typically used in DNP at high fields. Results from the simulations are shown, demonstrating the complex dependences of the DNP-assisted spin diffusion process on variations of the relevant parameters. In particular, the effect of the spin lattice relaxation times on the polarization buildup times and the resulting end polarization are discussed, and the quenching of the polarizations by the hyperfine interaction is demonstrated.     

(2)H-decoupling-accelerated (1)H spin diffusion in dynamic nuclear polarization with photoexcited triplet electrons

In dynamic nuclear polarization (DNP) experiments applied to organic solids for creating nonequilibrium, high (1)H spin polarization, an efficient buildup of (1)H polarization is attained by partially deuterating the material of interest with an appropriate (1)H concentration. In such a dilute (1)H spin system, it is shown that the (1)H spin diffusion rate and thereby the buildup efficiency of (1)H polarization can further be enhanced by continually applying radiofrequency irradiation for deuterium decoupling during the DNP process. As experimentally confirmed in this work, the electron spin polarization of the photoexcited triplet state is mainly transferred only to those (1)H spins, which are in the vicinity of the electron spins, and (1)H spin diffusion transports the localized (1)H polarization over the whole sample volume. The (1)H spin diffusion coefficients are estimated from DNP repetition interval dependence of the initial buildup rate of (1)H polarization, and the result indicates that the spin diffusion coefficient is enhanced by a factor of 2 compared to that without (2)H decoupling.     

Energy spectrum of extended Hubbard model with spin‐dependent hopping and related spin ladder model

We studied the energy spectrum of the 1‐D extended Hubbard model with spin‐dependent hopping and related spin ladder system formed by two coupled XXZ spin 1/2 chains with the interchain interaction of Ising type. It was proved that the model excitation spectrum has no gap excepting some special values of z‐projection of the ground‐state total spin. The thorough analytic consideration of two‐magnon states was given. The existence up to five bound states at specified value of quasimomentum of the pair of inverted spins was shown. We also present the results of density matrix renormalization group calculations that showed nonadequacy of the pair approximation for n‐magnon bound states of the extended model with the strong electron–electron interactions. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Time-dependent density functional theory/discrete reaction field spectra of open shell systems: The visual spectrum of [FeIII(PyPepS)2]- in aqueous solution

We report the calculated visible spectrum of [FeIII(PyPepS)2]- in aqueous solution. From all-classical molecular dynamics simulations on the solute and 200 water molecules with a polarizable force field, 25 solute/solvent configurations were chosen at random from a 50 ps production run and subjected the systems to calculations using time-dependent density functional theory (TD-DFT) for the solute, combined with a solvation model in which the water molecules carry charges and polarizabilities. In each calculation the first 60 excited states were collected in order to span the experimental spectrum. Since the solute has a doublet ground state several excitations to states are of type "three electrons in three orbitals," each of which gives rise to a manifold of a quartet and two doublet states which cannot properly be represented by single Slater determinants. We applied a tentative scheme to analyze this type of spin contamination in terms of Delta and Delta transitions between the same orbital pairs. Assuming the associated states as pure single determinants obtained from restricted calculations, we construct conformation state functions (CFSs), i.e., eigenfunctions of the Hamiltonian Sz and S2, for the two doublets and the quartet for each Delta,Delta pair, the necessary parameters coming from regular and spin-flip calculations. It appears that the lower final states remain where they were originally calculated, while the higher states move up by some tenths of an eV. In this case filtering out these higher states gives a spectrum that compares very well with experiment, but nevertheless we suggest investigating a possible (re)formulation of TD-DFT in terms of CFSs rather than determinants.     

Solvent effects on optically detected magnetic resonance in triplet spin labels

We have calculated solvent effects on the zero-field splitting (ZFS) constants induced by electron spin–spin coupling (SSC) in the low-lying triplet states of azaaromatic molecules in solutions using multiconfiguration self-consistent-field wave functions and the polarizable continuum model. The second-order spin–orbit coupling (SOC) contribution to the splitting of the ^3* states is found to be almost negligible, and the calculations therefore provide a good estimate of the ZFS parameters and their solvent dependence based only on the electron spin–spin coupling expectation values. The correlation between the shift in the ZFS and the phosphorescence frequency that has been observed in optically detected magnetic resonance experiments in low-temperature glasses is supported by our direct SSC calculations without taking SOC into account. This makes it possible to distinguish between the two theories that earlier were proposed to explain the inhomogeneous broadening of triplet state spectra, and discard the one that is exclusively based on the SOC-induced mixing of the singlet and triplet states.Contribution to the Jacopo Tomasi Honorary IssueAcknowledgments. This work was supported (B. M.) by the Swedish Royal Academy of Science (KVA). This work was also supported by the Norwegian Research Council through a grant of computer time from the Program for Supercomputing. We are grateful to B. Schimmelpfenning for his valuable assistance in the computations.     

Diagrammatic formulation of the second‐order many‐body multipartitioning perturbation theory

The second‐order multireference perturbation theory employing multiple partitioning of the many‐electron Hamiltonian into a zero‐order part and a perturbation is formulated in terms of many‐body diagrams. The essential difference from the standard diagrammatic technique of Hose and Kaldor concerns the rules of evaluation of energy denominators which take into account the dependence of the Hamiltonian partitioning on the bra and ket determinantal vectors of a given matrix element, as well as the presence of several two‐particle terms in zero‐order operators. The novel formulation naturally gives rise to a “sum‐over‐orbital” procedure of correlation calculations on molecular electronic states, particularly efficient in treating the problems with large number of correlated electrons and extensive one‐electron bases. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 395–401, 1999     

Critical assessment of electron spin resonance studies on Cu(I)-NO complexes in Cu-ZSM-5 zeolites prepared by solid- and liquid-state ion exchange

Cu(I)-NO adsorption complexes were formed over Cu-ZSM-5 zeolites prepared by (i) solid-state ion exchange of NH(4)-ZSM-5 with CuCl and (ii) liquid-state ion exchange of ZSM-5 with Cu(CH(3)COO)(2). Electron spin resonance spectroscopy revealed the formation of two different Cu(I)-NO species A and B in both systems, whose spin Hamiltonian parameters are comparable with those already reported for the Cu(I)-NO species formed over 66% Cu(II) liquid-state ion-exchanged Cu-ZSM-5 materials. The population of the species A and B differs for the two systems studied. Formation of species B is more favored in the solid-state ion-exchanged Cu-ZSM-5 when compared to the liquid-state exchanged zeolite. The X-, Q- and W-band electron spin resonance spectra recorded at 6 and 77 K reveal the presence of a rigid geometry of the adsorption complexes at 6 K and a dynamic complex structure at higher temperatures such as 77 K. This is indicated by the change in the spin Hamiltonian parameters of the formed Cu(I)-NO species in both the liquid- and solid-state ion-exchanged Cu-ZSM-5 zeolites from 6 to 77 K. Possible models for the motional effects found at elevated temperatures are discussed. The temperature dependence of the electron spin phase memory time measured by two-pulse electron spin-echo experiments indicates, likewise, the onset of a motional process of the adsorbed NO molecules at temperatures above 10 K. The studies support previous assignments where the NO complexes are formed at two different Cu(I) cationic sites in the ZSM-5 framework and highlight that multifrequency electron spin resonance experiments at low temperatures are essential for reliable determination of the spin Hamiltonian parameters of the formed adsorption complexes for further comparison with Cu(I)-NO complex structures predicted by quantum chemical calculations.     

Unpaired electrons at the second‐order reduced density matrix level: Covalent bonding,and coulomb and fermi correlations in closed shell systems

The usual one‐electron populations in atomic orbitals of closed shell systems are split into unpaired and paired at the (spin‐dependent) second‐order reduced density matrix level. The unpaired electron in an orbital is defined as the “simultaneous occurrence of an electron and an electron hole of opposite spins in the same spatial orbital,” which for simplicity is called “electropon.” The electropon population in a given orbital reveals whether and to what degree the Coulomb correlations, and hence, the chemical bonding between this orbital and the remaining orbitals of the system are globally favorable or unfavorable. The interaction of two electropons in two target orbitals reveals the quality (favorable or unfavorable) and the strength of the covalent bonding between these orbitals; this establish a bridge between the notion of “unpaired electrons” and the traditional covalent structure of valence‐bond (VB) theory. Favorable/unfavorable bonding between two orbitals is characterized by the positive/negative (Coulomb) correlation of two electropons of opposite spins, or alternatively, by the negative/positive (Fermi) correlation of two parallel spin electropons. A spin‐free index is defined, and the relationship between the electropon viewpoint for chemical bonding and the well‐known two‐electron Coulomb and Fermi correlations is established. Benchmark calculations are achieved for ethylene, hexatriene, benzene, pyrrole, methylamine, and ammonia molecules on the basis of physically meaningful natural orbitals. The results, obtained in the framework of both orthogonal and nonorthogonal population analysis methods, provide the same conceptual pictures, which are in very good agreement with elementary chemical knowledge and VB theory. © 2013 Wiley Periodicals, Inc.     

Numerical and experimental studies of long-range magnetic dipolar interactions

We describe several numerical methods developed to analyze the behavior of spin polarized liquids in the presence of long-range magnetic dipolar interactions and external field gradients. Two of the methods use a discrete lattice of spins. In the first we calculate the magnetic field from the lattice of spins directly, either in the rotating frame, or in the lab frame. In the second method we include the dipolar fields from linear magnetization gradients analytically and calculate the dipolar fields from higher order gradients in Fourier space, where they are a local function of the magnetization. In the third method the magnetization is expanded in a Taylor series and the dipolar fields are calculated analytically for each term. The results of these calculations are compared to experimental data, in which we use two superconducting quantum interference device magnetometers adjacent to a spherical sample of hyperpolarized liquid 129Xe to detect the evolution of magnetization gradients. In particular, we observe an increase by a factor of 100 of the spin dephasing time in a longitudinal magnetic field gradient due to dipolar interactions of the spins. While each of the numerical techniques has certain limitations, they are generally in agreement with each other and with experimental data.