Unrestricted Hartree-Fock calculations of spin densities with orthogonalized atomic orbitals: Proton and 13C hyperfine splittings in some pi hydrocarbon radicals

The spin density distribution in a few hydrocarbon radicals has been calculated using orthogonalized atomic orbitals in the Unrestricted Hartree-Fock formalism of Amos and Snyder and including certain more important two-electron hybrid and exchange integrals and all the core-resonance integrals. Our calculated spin densities for the cation and anion radicals of alternant hydrocarbons, which are now different due to the breakdown of the pairing theorem, are, in general, of the right relative order so that even the simple McConnell type of relation can account partly for the observed differences in the proton splittings between cations and anions. The proton splittings for position 2 of naphthalene and anthracene radical ions are correctly predicted, thus clearing up the well-known cation-anion anomaly for this position. Comparative calculations have been made to show that the spin density results are worsened with the neglect of the integrals of the type mentioned before. An empirical analysis correlating the observed ¹³C splittings and the spin density results over a non-orthogonal basis set shows that the available ¹³C splittings in alternant hydrocarbon radical ions can be explained with a set of sigma-pi parameters which are consistent with the theory. It is shown that even though the spin densities in cations and anions may be different, these can lead to similar ¹³C splittings.     

Aminyloxide (nitroxide)-XXVI: ESR-spektroskopische untersuchung von hydrazonyloxiden. Ermittlung der spindichteverteilung

The so-called “Bergmann oxide” 4a and the related compounds 4b-i dissociate reversibly to the corresponding radicals 5 at elevated temperatures. Analysis of the ESR spectra reveals that in 5a-f the unpaired electron is delocalized over the entire molecule. On the other hand the strongly reduced spin density at the β-carbon atom in 5h-i as well as in 7 indicates a twisting about the N,N-bond in these radicals, whereas m 5g the bond between the β-carbon atom and the group Ar¹ is twisted. The results of spin density calculations for radicals 5 are in agreement with the experimentally estimated spin densities. In spin trap experiments with nitrosobenzene or nitrones respectively, 5a reacts at the β-carbon atom, indicating this position as the most reactive one.     

Insights on spin polarization through the spin density source function

Understanding how spin information is transmitted from paramagnetic to non-magnetic centers is crucial in advanced materials research and calls for novel interpretive tools. Herein, we show that the spin density at a point may be seen as determined by a local source function for such density, operating at all other points of space. Integration of the local source over Bader''s quantum atoms measures their contribution in determining the spin polarization at any system''s location. Each contribution may be then conveniently decomposed in a magnetic term due to the magnetic natural orbital(s) density and in a reaction or relaxation term due to the remaining orbitals density. A simple test case, ³B₁ water, is chosen to exemplify whether an atom or group of atoms concur or oppose the paramagnetic center in determining a given local spin polarization. Discriminating magnetic from reaction or relaxation contributions to such behaviour strongly enhances chemical insight, though care needs to be paid to the large sensitivity of the latter contributions to the level of the computational approach and to the difficulty of singling out the magnetic orbitals in the case of highly correlated systems. Comparison of source function atomic contributions to the spin density with those reconstructing the electron density at a system''s position, enlightens how the mechanisms which determine the two densities may in general differ and how diverse may be the role played by each system''s atom in determining each of the two densities. These mechanisms reflect the quite diverse portraits of the electron density and electron spin density Laplacians, hence the different local concentration/dilution of the total and (α–β) electron densities throughout the system. Being defined in terms of an observable, the source function for the spin density is also potentially amenable to experimental determination, as customarily performed for its electron density analogue.     

Electron spin density distribution of a series of nitroxide radicals with five-member ring as studied by solid-state 1H, 2H and 13C NMR

Stable nitroxide radicals are useful to construct molecular magnetic systems. Particularly, radicals substituted by –COOH and –CONH₂ can be coordinated to magnetic metal ions and may be used as cladding reagents of gold nano-particles for modifying magnetism. Nitroxide molecules with unsaturated five-member ring have almost planner structure and electron spin delocalization may be expected. We determined the hyperfine coupling constants (hfcc) of ¹H, ²H and ¹³C of a series of nitroxide radicals with five-member ring. Experimental values of hfcc were compared with those deduced from calculations based on density functional theory. The electron spin density distribution at β position of ring was sensitive to the ring structure, although the electron spin density at β position is small compared with N–O site. Magnetic susceptibility and UV–Vis absorption spectra were also measured and discussed.     

Application of the unitary group approach to evaluate spin density for configuration interaction calculations in a basis of S2 eigenfunctions

We present an implementation of the spin‐dependent unitary group approach to calculate spin densities for configuration interaction calculations in a basis of spin symmetry‐adapted functions. Using S² eigenfunctions helps to reduce the size of configuration space and is beneficial in studies of the systems where selection of states of specific spin symmetry is crucial. To achieve this, we combine the method to calculate U(n) generator matrix elements developed by Downward and Robb (Theor. Chim. Acta 1977, 46, 129) with the approach of Battle and Gould to calculate U(2n) generator matrix elements (Chem. Phys. Lett. 1993, 201, 284). We also compare and contrast the spin density formulated in terms of the spin‐independent unitary generators arising from the group theory formalism and equivalent formulation of the spin density representation in terms of the one‐ and two‐electron charge densities.     

Polarized neutron diffraction and its application to spin density studies

Spin density distributions in molecular compounds containing unpaired electrons have been studied by polarized neutron diffraction (PND). The spin density distributions provide a unique perspective of the magnetic properties of the compounds. The background and fundamentals of polarized neutron diffraction are summarized in this review, followed by examples of applications in inorganic and organic chemistry. Spin densities in several compounds that are obtained by polarized neutron diffraction are highlighted. Spin densities in single molecular magnet [Fe₈O₂(OH)₁₂(tacn)₆]⁸⁺ and cyano-bridged K₂[Mn(H₂O)₂]₃[Mo(CN)₇]₂·6H₂O demonstrate how to obtain magnetic interaction in the complexes by PND. PND studies of Ru(acac)₃, containing one single unpaired electron, show small spin densities in this complex. Finally the use of PND in studying nitronyl nitroxide radicals is given. Our goal in this review is to illustrate how PND functions and how it serves as a sensitive tool in directly probing spin density in molecules.     

Isotropic paramagnetic shifts of the phosphorus and proton magnetic resonance and the spin density distribution in paramagnetic complexes of triphenylphosphine with nickel acetylacetonate

The Isotropic paramagnetic shifts in the H¹ and P³¹ NMR spectra of triphenylphosphine, forming labile complexes in chloroform with the acetylacetonato compound of nickel(II), have been measured. The concentration dependence of the shifts has been used to calculate the hyperfine interaction constants and the spin densities in an individual complex, on the assumption that the complex has the composition [Ni(AA)₂] (PPh₃)₂. It has been found that the spin density in the sp³ hybrid orbital of the phosphorus atom is positive and amounts to not less than 10%, indicating a -mechanism of transfer of spin density from the nickel to the phosphorus. The spin densities on the carbon atoms of the phenyl rings are smaller fay about two orders of magnitude than those on the phosphorus atom, and this is attributed to the orientation of these rings, which is unsuitable for conjugation with the unshared pair. The contributions of the different mechanisms to the delocalization of the spin density from the phosphorus atom to the -system of the rings have been estimated.The authors wish to express their gratitude to the late Academician V. V. Voevodskii, whose initiative and constant support made this joint work possible. The authors thank A. Nesmei, for assistance in the construction of the phosphorus resonance measurement apparatus, and G. M. Zhidomirov, for discussion of the results.     

On-top pair-density interpretation of spin density functional theory,with applications to magnetism

The on-top pair density P(r, r) gives the probability that one electron will be found on top of another at position r. We find that the local spin density (LSD) and generalized gradient (GGA) approximations for exchange and correlation predict this quantity with remarkable accuracy. We show how this fact and the usual sum-rule arguments explain the success of these approximations for real atoms, molecules, and solids, where the electron spin densities do not vary slowly over space. Self-consistent LSD or GGA calculations make realistic predictions for the total energy E, the total density n(r), and the on-top pair density P(r,r), even in those strongly “abnormal” systems (such as stretched H₂) where these approximations break symmetries and yield unrealistic spin magnetization densities m(r). We then suggest that ground-state ferromagnetic iron is a “normal” system, for which for LSD or GGA m(r) and the related local spin moment are trustworthy, but that iron above the Curie temperature and antiferromagnetic clusters at all temperatures are abnormal system for which the on-top pair density interpretation is more viable than the standard physical interpretation. As an example of a weakly abnormal system, we consider the four-electron ion with nuclear charge Z → ∞ © 1997 John Wiley & Sons, Inc.     

Quantum Chemical Spin Densities for Radical Cations of Photosynthetic Pigment Models

The spin densities of radical cations of magnesium porphyrin, magnesium chlorine and a truncated chlorophyll a model are calculated with density‐functional theory and multiconfigurational quantum chemical methods. The latter serve as a reference for approximate density‐functional theory which yields spin densities that may suffer from the self‐interaction error. We carried out complete active space self‐consistent field calculations with increasing active orbital spaces to systematically converge qualitatively correct spin densities. In particular, for the magnesium chlorine and chlorophyll a model radical cations, this is not easy to achieve because of the lower symmetry compared to magnesium porphyrin. Strategies had to be employed which allowed us to consider very large active orbital spaces. We explored restricted active space self‐consistent field and density‐matrix renormalization group calculations. Based on these reference data, we assessed the accuracy of different density‐functional approximations. We show that in particular, exchange–correlation model potentials with correct asymptotic behavior yield good spin densities, and we find, in agreement with previous studies on different classes of compounds, that hybrid functionals systematically increase spin‐polarization effects with increasing amounts of exact exchange. Our results provide a starting point for investigations of spin densities of more complex systems such as the hinge model for the primary electron donor in photosystem II.     

Determination of the nuclear structure and spin density distribution in the cyano-bridged molecular based magnet K2Mn3(H2O)6[Mo(CN)7]2·6 H2O

This article details the local spin density determination in the cyano-bridged, two-dimensional, molecular based magnet K₂Mn₃(H₂O)₆[Mo(CN)₇]₂·6 H₂O (T_c = 39 K). The crystal structure, determined at room temperature by X-ray diffraction, was redetermined at 50 K using unpolarised neutron diffraction. The importance of intermolecular hydrogen-bonding interactions is clearly demonstrated in this study, previously characterised with X-rays. The local spin density was determined from polarised neutron diffraction data at 4 K with an applied field of 3 T. Positive spin densities were observed on the manganese sites, consistent with high-spin d⁵ ions in octahedral fields, whilst a negative spin density was found on the molybdenum sites, signifying delocalisation onto the cyano ligands. The alternating sign of the spin populations on the metal sites, suggests that the primary Mo^III–Mn^II interactions are antiferromagnetic in nature and the delocalisation onto the cyano-bridges clearly demonstrates the role of the ligand bridges in the magnetic exchange pathway.     

DFT-BS study on the magnetic coupling interaction in a series of (R-Bpmp)Mn2(μ-OAc)2 complexes

The magnetic properties of a series of dinuclear Mn^II systems are investigated by the calculations based on density functional theory combined with broken-symmetry approach (DEF-BS). It is found that there are weak antiferromagnetic interactions in these systems with different bridging ligands. The changing trend of the magnetic coupling constants J indicates that with the electronegativity of the increasing bridging ligands, the antiferromagnetic coupling interaction is weakened. The analyses of the magnetic orbitals and the spin densities show that the weakly antiferromagnetic couplings in these systems are due to the vertical magnetic d orbitals and the weak spin delocalization. These results should be instructive for the design of new molecular magnetic materials.     

Density matrices from position and momentum densities

Summary One-electron density matrices, which are representable in single-centers-orbital basis sets, have been investigated with respect to their reconstruction from densities. The maximum allowed dimension for reconstruction from a combination of position & momentum density dependent properties is only slightly bigger than the dimension in the case of position (or momentum) densities only. Since for a given one-particle basis of dimensionM, the number of one-matrix elements which can be determined is also of orderM only, while the total number of one-matrix elements is of orderM ², it is in general necessary to introduce severe constraints and restrictions. The accuracy demands on the data and algorithms increase exponentially for linearly increasing size of basis set.     

Pseudocontact contribution to the isotropic chemical shifts of protons in octahedral paramagnetic complex ions

Following the formalism of Kurland and McGarvey, it is shown that transferred spin densities onto ligands can give rise to pseudocontact shifts even for protons in octahedral complexes. In the case of ammonia as a ligand the pseudocontact shift of the protons originates from the spin density on the p orbitals of the nitrogen as well as spin density on the s orbitals on the neighboring hydrogen atoms. Using LCAO-MO wavefunctions for the Ru(NH₃)³⁺₆ ion the contact and pseudocontact shifts were estimated to contribute about 75% and 25% of the total shift of the amine protons respectively. Based on experimental paramagnetic NMR shifts, the covalency parameter in the Ru(NH₃)³⁺₆ ion was recalculated to be λ = 0.325.     

Pulsed EPR and NMR spectroscopy of paramagnetic iron porphyrinates and related iron macrocycles: how to understand patterns of spin delocalization and recognize macrocycle radicals

Pulsed EPR spectroscopic techniques, including ESEEM (electron spin echo envelope modulation) and pulsed ENDOR (electron-nuclear double resonance), are extremely useful for determining the magnitudes of the hyperfine couplings of macrocycle and axial ligand nuclei to the unpaired electron(s) on the metal as a function of magnetic field orientation relative to the complex. These data can frequently be used to determine the orientation of the g-tensor and the distribution of spin density over the macrocycle, and to determine the metal orbital(s) containing unpaired electrons and the macrocycle orbital(s) involved in spin delocalization. However, these studies cannot be carried out on metal complexes that do not have resolved EPR signals, as in the case of paramagnetic even-electron metal complexes. In addition, the signs of the hyperfine couplings, which are not determined directly in either ESEEM or pulsed ENDOR experiments, are often needed in order to translate hyperfine couplings into spin densities. In these cases, NMR isotropic (hyperfine) shifts are extremely useful in determining the amount and sign of the spin density at each nucleus probed. For metal complexes of aromatic macrocycles such as porphyrins, chlorins, or corroles, simple rules allow prediction of whether spin delocalization occurs through sigma or pi bonds, and whether spin density on the ligands is of the same or opposite sign as that on the metal. In cases where the amount of spin density on the macrocycle and axial ligands is found to be too large for simple metal-ligand spin delocalization, a macrocycle radical may be suspected. Large spin density on the macrocycle that is of the same sign as that on the metal provides clear evidence of either no coupling or weak ferromagnetic coupling of a macrocycle radical to the unpaired electron(s) on the metal, while large spin density on the macrocycle that is of opposite sign to that on the metal provides clear evidence of antiferromagnetic coupling. The latter is found in a few iron porphyrinates and in most iron corrolates that have been reported thus far. It is now clear that iron corrolates are remarkably noninnocent complexes, with both negative and positive spin density on the macrocycle: for all chloroiron corrolates reported thus far, the balance of positive and negative spin density yields -0.65 to -0.79 spin on the macrocycle. On the other hand, for phenyliron corrolates, the balance of spin density on the macrocycle is zero, to within the accuracy of the calculations (Zakharieva, O.; Schünemann, V.; Gerdan, M.; Licoccia, S.; Cai, S.; Walker, F. A.; Trautwein, A. X. J. Am. Chem. Soc. 2002, 124, 6636-6648), although both negative and positive spin densities are found on the individual atoms. DFT calculations are invaluable in providing calculated spin densities at positions that can be probed by (1)H NMR spectroscopy, and the good agreement between calculated spin densities and measured hyperfine shifts at these positions leads to increased confidence in the calculated spin densities at positions that cannot be directly probed by (1)H NMR spectroscopy. (13)C NMR spectroscopic investigations of these complexes should be carried out to probe experimentally the nonprotonated carbon spin densities.     

Temperature Dependence of Interactions between Stable Piperidine‐1‐yloxyl Derivatives and an Ionic Liquid

2,2,6,6‐Tetramethylpiperidine‐1‐yloxyl derivatives substituted with either hydrogen bonding [‐OH, ‐OSO₃H] or ionic [‐OSO₃^?Na⁺, ‐OSO₃^?K⁺, N⁺(CH₃)₃I^?, N⁺(CH₃)₃ N^?(SO₂‐CF₃)₂] substituents are investigated in 1‐butyl‐3‐methylimidazolium tetrafluoroborate over a wide temperature range covering both glassy and viscous states. The Vogel–Fulcher–Tammann equation describes the temperature dependence of the ionic liquid viscosity. Quantum chemical calculations of the spin probes at the UB3LYP/6‐311(2d,p++) level are done to describe the dependence of the spin density on nitrogen on the substitution pattern of the 4‐position of the probe. The results of these calculations are also used to understand the experimental results obtained by applying the Spernol–Gierer–Wirtz theory to analyze the viscosity dependence of the rotational correlation time of the spin probes. Significant differences are found between 2,2,6,6‐tetramethylpiperidine‐1‐yloxyl and its derivatives containing substituents that are able to form hydrogen bonds with the ionic liquid. Moreover, derivatives substituted with ionic groups at the 4‐position have a large effect on temperature‐induced solvent viscosity, as this is particularly dependent on the nature of the substituent at the 4‐position. These dependencies include the temperature region that can be used to describe interactions between the spin probes and the ionic liquid, diffusion into the free volume during non‐activated (neutral spin probes) and activated (charged spin probes) processes. Additional parameters are the radii of the ionic liquid and the spin probes, which are calculated and measured approximately. In addition, the temperature dependence of the isotropic hyperfine coupling constants of the spin probes results in information about the micropolarity of the ionic liquid. At room temperature, this is comparable to that of the solvent dimethylsulfoxide.     

Spin–Orbit Contributions in High‐Spin Nitrenes/Carbenes: A Hybrid CASSCF/MRMP2 Study of Zero‐Field Splitting Tensors

Zero‐field splitting (ZFS) tensors ( D tensors) of organic high‐spin oligonitrenes/oligocarbenes up to spin‐septet are quantitatively determined on the basis of quantum chemical calculations. The spin–orbit contributions, D ^SO tensors are calculated in terms of a hybrid CASSCF/MRMP2 approach, which was recently proposed by us. The spin–spin counterparts, D ^SS tensors are computed based on McWeeny–Mizuno’s equation in conjunction with the RODFT spin densities. The present calculations show that more than 10 % of ZFS arises from spin–orbit interactions in the high‐spin nitrenes under study. Contributions of spin‐bearing site–site interactions are estimated with the aid of a semi‐empirical model for the D tensors and found to be ca. 5 % of the D ^SO tensor. The analysis of intermediate states reveal that the largest contributions to the calculated D ^SO tensors are attributed to intra‐site spin flip excitations and delocalized π and π* orbitals play an important role in the inter‐site spin–orbit interactions.     

Discrepancy between the Spin Distribution and the Magnetic Ground State for a Triaminoxyl Substituted Triphenylphosphine Oxide Derivative

The magnetic interaction and spin transfer via phosphorus have been investigated for the tri‐tert‐butylaminoxyl para‐substituted triphenylphosphine oxide. For this radical unit, the conjugation existing between the π* orbital of the NO group and the phenyl π orbitals leads to an efficient delocalization of the spin from the radical to the neighboring aromatic ring. This has been confirmed by using fluid solution high‐resolution EPR and solid state MAS NMR spectroscopy. The spin densities located on the atoms of the molecule could be probed since ¹H, ¹³C, ¹⁴N, and ³¹P are nuclei active in NMR and EPR, and lead to a precise spin distribution map for the triradical. The experimental investigations were completed by a DFT computational study. These techniques established in particular that spin density is located at the phosphorus (ρ=?15×10^?3 au), that its sign is in line with the sign alternation principle and that its magnitude is in the order of that found on the aromatic C atoms of the molecule. Surprisingly, whereas the spin distribution scheme supports ferromagnetic interactions among the radical units, the magnetic behavior found for this molecule revealed a low‐spin ground state characterized by an intramolecular exchange parameter of J=?7.55 cm^?1 as revealed by solid state susceptibility studies and low temperature EPR. The X‐ray crystal structures solved at 293 and 30 K show the occurrence of a crystallographic transition resulting in an ordering of the molecular units at low temperature.     

Theoretical approach to fluorine substitution in X2NO and X2CN free radicals. Comparison between ab initio UHF and RHF + perturbation treatments

Non-empirical calculations have been performed to analyze the effects of fluorine substitution on the geometry and electronic properties of two series of π and σ radicals. Both UHF and RHF + perturbation methods have been used and the results are compared as a function of the basis set quality. As concerns geometry and spin-free electronic properties the results are independent of the UHF or RHF formalism, but highly sensitive to the basis set. The STO-3G basis is unable to reproduce even general trends. Polarization functions always play a relevant role and correlation effects seem not negligible at least for fluorine-containing radicals. The molecular shape of π radicals changes from a planar to a pyramidal geometry when increasing the electronegativity of the substituents. On the contrary, σ radicals always remain planar. Unprojected UHF spin densities are closer to the RHF + perturbation results for small spin contamination (X₂NO). On the contrary, it is the projected UHF spin density which is in better agreement with the RHF + perturbation value for large spin contamination (X₂CN). No simple correlation can be found between spin densities and gross atomic spin populations. For H₂NO the spin density at nitrogen is smaller than at the oxygen nucleus, but substitution may enhance or reverse this trend.