Theoretical study of magnetic properties of ammonia molecule in nonuniform magnetic field

The interaction Hamiltonian within the Bloch gauge for the potentials of the electromagnetic field has been used to define magnetic multipole moment operators and operators for the magnetic field of electrons acting on the nuclei of a molecule in the presence of nonhomogeneous external magnetic field. Perturbation theory has been applied to evaluate the induced electronic moments and magnetic field at the nuclei. Multipole magnetic susceptibility and nuclear magnetic shielding tensors have been introduced to describe the contributions arising in nonuniform fields, and their origin dependence has been analyzed. Extended numerical tests on the ammonia molecule in a static, nonuniform magnetic field have been carried out, using the random-phase approximation within the framework of accurate Hartree-Fock zero-order wavefunctions, and allowing for both angular momentum and torque formalisms in the calculation of paramagnetic contributions.     

Ab initio calculation of magnetic properties by the “direct” IGLO method

A direct method for the ab initio calculation of the magnetic susceptibility and chemical shielding tensors based on the individual gauge for localized molecular orbitals (IGLO) formalism is introduced. “Direct” in this context means we avoid storing the two-electron repulsion integrals in favor of recalculating them whenever necessary. In conjunction with the Direct-SCF package TURBOMOLE Direct IGLO (DIGLO) permits calculation of magnetic second-order properties for large molecules by minimizing peripheral disc storage requirements. The size of the molecules to be treated is limited only by the amount of CPU time available. The performance of DIGLO is demonstrated for some selected examples.     

On the existence of a natural common gauge-origin for the calculation of magnetic properties of atoms and molecules via gaugeless basis sets

It is proven that, within the conventional approach using a common origin and gaugeless basis sets for the calculation of atomic magnetizability and Larmor current density induced by an external magnetic field, the natural gauge origin coincides with the nucleus. Recipes for defining an optimal gauge origin for the calculation of magnetizability and magnetic shielding at the nuclei of a molecule are given. Within the common origin approach, the paramagnetic contributions to the components of magnetic tensors of a molecule are represented by a minimum number of non-vanishing parameters if the gauge origin is chosen at a point characterized by the total molecular symmetry, e.g., the center of electronic charge for magnetizabilities. It is shown that total values of diagonal components of the magnetic shielding tensor σ(I) at a nucleus I in a molecule, as well as separate diamagnetic σ(dI) and paramagnetic σ(pI) contributions, calculated via the common origin method, are origin independent for a number of local point group symmetries. The diagonal components (and the average value) of σ(I) depend on the gauge origin only for nuclear site symmetries C(1), C(s), C(n), C(nv), n = 2, 3.... Group-theoretical methods show interesting features, e.g., for S(4) local symmetry, in a coordinate transformation, the paramagnetic contribution to the zz component and to the trace of the shielding tensor is origin independent, whereas the xx and yy components mix into one another, in such a way that their sum remains constant.     

Reduced first order density matrix for the Be ground state

A reduced first order density matrix for the Be ground state is computed from an extensive configuration interaction (CI ) wave function. A sequence of increasingly accurate CI wave functions Φ_q converging towards the exact Ψ is used to assess the quality of the results which include approximate bounds for the overlaps 〈Φ_q|Ψ〉, electron–nuclear coalescence cusp data, Weinhold's overlap between density matrices, virial ratios, occupation number spectra, and some expectation values. The nuclear magnetic shielding constant and the molar diamagnetic susceptibility are determined with 2.0 and 1.5% of uncertainty, respectively.     

An Investigation of Lanthanum Coordination Compounds by Using Solid‐State 139La NMR Spectroscopy and Relativistic Density Functional Theory

Lanthanum‐139 NMR spectra of stationary samples of several solid La^III coordination compounds have been obtained at applied magnetic fields of 11.75 and 17.60 T. The breadth and shape of the ¹³⁹La NMR spectra of the central transition are dominated by the interaction between the ¹³⁹La nuclear quadrupole moment and the electric field gradient (EFG) at that nucleus; however, the influence of chemical‐shift anisotropy on the NMR spectra is non‐negligible for the majority of the compounds investigated. Analysis of the experimental NMR spectra reveals that the ¹³⁹La quadrupolar coupling constants (C_Q) range from 10.0 to 35.6 MHz, the spans of the chemical‐shift tensor (Ω) range from 50 to 260 ppm, and the isotropic chemical shifts (δ_iso) range from ?80 to 178 ppm. In general, there is a correlation between the magnitudes of C_Q and Ω, and δ_iso is shown to depend on the La coordination number. Magnetic‐shielding tensors, calculated by using relativistic zeroth‐order regular approximation density functional theory (ZORA‐DFT) and incorporating scalar only or scalar plus spin–orbit relativistic effects, qualitatively reproduce the experimental chemical‐shift tensors. In general, the inclusion of spin–orbit coupling yields results that are in better agreement with those from the experiment. The magnetic‐shielding calculations and experimentally determined Euler angles can be used to predict the orientation of the chemical‐shift and EFG tensors in the molecular frame. This study demonstrates that solid‐state ¹³⁹La NMR spectroscopy is a useful characterization method and can provide insight into the molecular structure of lanthanum coordination compounds.     

Electron correlation effects on magnetic properties of BH

The nuclear magnetic shielding and magnetizability tensors for the BH molecule are calculated by the coupled-Hartree–Fock method and many-body perturbation theory. As in the case of H₂, HF, and F₂, the second-order non-CHF diagrams make an inappreciable contribution.     

Correlated and gauge origin independent calculations of magnetic properties

Summary We have applied a gauge origin invariant method for calculations of nuclear magnetic shielding constants to the singly bonded molecules BF, F₂, BH₃, CH₄, NH₃, H₂O, and HF as well as to the¹H shielding constants of HCN and C₂H₂. The calculations were performed at the RPA and second order polarization propagator (SOPPA) level of theory. For most molecules the correlation contribution in SOPPA is less diamagnetic than in the comparable MP2 calculations. For F₂, SOPPA gives a large paramagnetic correlation correction whereas the MP2 method gives a very small correlation contribution. For all molecules agreement with experimental results is generally improved at the SOPPA level compared to RPA. We have also demonstrated that second order gauge origin invariant, common and local origin (SOLO) methods do not necessarily give the same shielding even in the limit of a converged basis set.     

Gauge invariance of the nuclear spin/electron orbit interaction and NMR spectral parameters

A gauge transformation of the vector potential A(m(I) ), associated to the magnetic dipole m(I) of nucleus I in a molecule, has been studied. The conditions for gauge invariance of nuclear magnetic shielding, nuclear spin/electron orbit contribution to spin-spin coupling between two nuclei, I and J, and electronic current density induced by m(I), have been expressed via quantum mechanical sum rules that are identically satisfied for exact and optimal variational wavefunctions. It is shown that separate diamagnetic and paramagnetic contributions to the properties transform into one another in the gauge transformation, whereas their sum is invariant. Therefore, only total response properties have a physical meaning. In particular, the disjoint diamagnetic and paramagnetic components of nuclear spin/electron orbit contributions to coupling constants are not uniquely defined. The diamagnetic contribution to the nuclear spin-spin coupling tensor, evaluated as an expectation value in the Ramsey theory, can alternatively be expressed as a sum-over-states formula, by rewriting the second-order Hamiltonian in commutator form a? la Geertsen, as previously reported by Sauer. Other sum-over-states formulae are obtained via a gauge transformation, by a procedure formally allowing for a continuous translation of the origin of the m(I)-induced current density, analogous to those previously proposed for magnetizabilities and nuclear magnetic shielding.     

13C carbonyl chemical shielding tensors: Comparing SCF,MBPT (2), and DFT predictions to experiment

In this work, we calculate the ¹³C nuclear magnetic resonance chemical shielding tensors for 18 carbonyl-containing compounds. The many-body perturbation theory (MBPT), self-consistent field (SCF), and density functional theory (DFT) formalisms were used with gauge including atomic orbitals (GIAO) to calculate the shielding tensors. Our data suggest that shielding tensors can be efficiently estimated by performing one MBPT(2) correlated calculation (e.g., at a reference geometry) and SCF-level calculations at other geometries and taking the SCF-to-correlated tensor element differences to be geometry independent. That is, the correlation contribution to the chemical shielding seems to be relatively constant over a considerable range of distortions. Treatment of correlation using DFT methods is shown to not be as systematically reliable as with MBPT(2). Data on 18 carbonyl compounds show that the single largest influence on the shielding tensor is the presence of nearby electron-withdrawing or electron-donating groups. Finally, although good agreement with powder or single-crystal experimental data is achieved for two or three tensor eigenvalues, systematic differences remain for one element; the origins of these differences are discussed. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 875–894, 1997     

Some mathematical properties of gauge transformations with respect to the Coulomb gauge: Variational analysis of an energy functional

As gauge invariance of computed magnetic properties, usually partitioned into diamagnetic and paramagnetic terms, is not achieved within the algebraic approximation, unless ad hoc techniques are adopted, a general variational treatment is analyzed, attempting to minimize the term more difficult to evaluate accurately, i.e., the paramagnetic contribution to magnetic susceptibility, by means of a gauge transformation. It is shown that an absolute minimum in a variational sense cannot be determined a priori. However, a “local” minimum of the paramagnetic contribution to magnetic susceptibility can be arrived at by employing general gauge transformations of polynomial form. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 599–606, 2000     

Comparison of Magnetic Properties between Spin Singlet and Triplet Li<Subscript>3</Subscript>Al<Stack><Subscript>4</Subscript><Superscript>−</Superscript></Stack> Clusters

With a second-order Møller–Plesset perturbation theory and Hartree–Fock nuclear magnetic resonance calculations, we investigated the magnetic properties of spin singlet and triplet Li₃Al ₄ ^? clusters. The obtained gauge-independent atomic orbital magnetic shielding tensors confirm the paramagnetism of singlet Li₃Al ₄ ^? and diamagnetism of the triplet. The planar rings composed of four aluminum atoms make the magnetic properties of Li₃Al ₄ ^? clusters versatile. The localized molecular orbital, low symmetry of geometric conformation and narrow gap between the highest occupied molecular orbital and the lowest unoccupied molecular orbital are found to correlate with the paramagnetism of singlet Li₃Al ₄ ^? . The origin of the paramagnetism is explained. In triplet Li₃Al ₄ ^? , the two outmost orbitals are degenerate, causing a conversion from the paramagnetism to diamagnetism.     

Eichinvariante Berechnung des Diamagnetismus und der chemischen Verschiebung

The procedure derived in^1–6 was used to calculate the magnetic susceptibilities and the magnetic shielding of a series of diamagnetic molecules (CH₄, C₂H₆, C₂H₄, C₂H₂, and HCN). The model requires only the knowledge of the unperturbed electron density. An incrementary treatment of the susceptibility-and shielding tensors is based on a partition of the electron density into localized parts.The values obtained by this procedure are in good agreement with experimental results in the case of susceptibilities and magnetic protonshieldings. The results for the¹³C-shielding however are not satisfactory.

     

Calculations of the magnetic shielding constants of heavy nuclei in polyatomic molecules

Coupled Hartree-Fock perturbation theory has been applied to compute the nuclear magnetic shielding tensors for ¹⁷O, ¹⁴N, and ¹³C in the molecules of water, ammonia, and methane with four wave functions of increasing accuracy, expanded over basis sets of Gaussian functions. The agreement with the experimental data available for ammonia and methane is very good. Quantities necessary to evaluate the shilding for an arbitrary gauge are also given. The degree of gauge-invariance of the calculated properties is nt satisfactory in the ammonia, while better results are found for water.     

A Linear 3d–4f Tetranuclear CoIII2DyIII2 Single‐Molecule Magnet: Synthesis,Structure, and Magnetic Properties

A linear tetranuclear 3d–4f Co₂Dy₂ cluster assembled from a polydentate Schiff base exhibits single‐molecule magnet (SMM) behavior with an anisotropic barrier of 33.8 K. Due to the presence of diamagnetic cobalt(III) ions, the tetranuclear cluster of 1 behaves magnetically like a dinuclear Dy₂ system. However, the diamagnetic segment might efficiently minimize undesirable intermolecular magnetic interactions, thereby improving the performance of the SMM behavior of 1 . This discrete complex presents us with a unique opportunity to study the magnetic properties and to probe the dynamics of magnetization in a magnetically isolated Dy₂ system.     

Ab initio calculation of magnetic susceptibilities and screening constants using gauge invariant Gaussian orbitals

The magnetic susceptibility and screening constant tensors are calculated using an ab initio finite perturbation SCF method, with gauge invariant Gaussian orbitals. The isoelectronic BH BeH^?, and CH⁺ molecules have been studied. The calculated values for BH are relatively different from those obtained by other methods. The CH⁺ molecule seems to exhibit a stronger temperature independent paramagnetism than BH, while the BeH^? molecule would be diamagnetic. The screening constants of the heavy atom of these molecules present a similar variation.     

On CHF calculations of second-order magnetic properties using the method of continuous transformation of origin of the current density

Summary A fully analytical formulation is outlined for computing molecular magnetic susceptibilities and nuclear magnetic shieldings via a continuous change of origin of the electronic current density induced by an external magnetic field. The change of origin is described in terms of a (continuous) arbitrary shift functiond(r). Coupled Hartree-Fock second-order magnetic properties of CH₄ and CO₂ molecules have been computed, using the special choiced(r)=r as generating function. A detailed analysis of results obtained with a variety of basis sets reveals that such a method is not as good as previously suggested. Large basis sets must be used to obtain accurate magnetic properties. On the other hand, all the components of theoretical nuclear magnetic shielding evaluated via this approach are independent of the origin of the vector potential. In general, theoretical magnetic susceptibilities depend linearly on the distance between different coordinate frames, but are origin independent for centre-symmetric molecules.     

Proton magnetic shielding in malonic acid by multiple pulse techniques

The proton NMR in single crystals of malonic acid has been studied by multiple pulse line narrowing techniques. The nuclear magnetic shielding tensors σ⁽ⁱ⁾ of all protons in malonic acid could be determined from the spectra. There are two magnetically distinct carboxyl protons. The principal components of their shielding tensors are found to be σ⁽¹⁾_ZZ= ?0.8 ppm, σ⁽¹⁾_YY = ?19.2 ppm, σ⁽¹⁾_XX = ?21.8 ppm, and σ⁽²⁾_ZZ = ?1.0 ppm, σ⁽²⁾_ZZ = ?21.3 ppm relative to adamantane. The error limits are estimated to be ± 1 ppm. The most shielded directions lie along the hydrogen bond directions to within 8 degrees. The least shielded directions are essentially perpendicular to the plane of the carboxyl groups. Within experimental accuracy the shielding of the aliphatic protons is axially symmetric about the CH bond axes. The anisotropy Δσ = σ_? ? σ_⊥ is (4 ± 1) ppm. The gross features of the anisotropy of the carboxyl protons are shown to be governed by the diamagnetic effect.     

Polarized Neutron Diffraction to Probe Local Magnetic Anisotropy of a Low‐Spin Fe(III) Complex

We have determined by polarized neutron diffraction (PND) the low‐temperature molecular magnetic susceptibility tensor of the anisotropic low‐spin complex PPh₄[Fe^III(Tp)(CN)₃]?H₂O. We found the existence of a pronounced molecular easy magnetization axis, almost parallel to the C₃ pseudo‐axis of the molecule, which also corresponds to a trigonal elongation direction of the octahedral coordination sphere of the Fe^III ion. The PND results are coherent with electron paramagnetic resonance (EPR) spectroscopy, magnetometry, and ab initio investigations. Through this particular example, we demonstrate the capabilities of PND to provide a unique, direct, and straightforward picture of the magnetic anisotropy and susceptibility tensors, offering a clear‐cut way to establish magneto‐structural correlations in paramagnetic molecular complexes.     

Molecular orbital studies of the NMR hydrogen bond shift

Magnetic shielding constants are calculated for the protons in XOH and XOH…OH₂ (XH, CH₃, NH₂, OH and F) molecules using a slightly extended set of atomic functions modified by gauge factors. These results are used to determine theoretical values for the NMR hydrogen bond shifts in the XOH…OH₂ systems. Such theoretical data are consistent with the few available experimental data. An analysis of the theoretical results reveals that there are three major types of shielding contribution to the NMR hydrogen bond shift; (a) a deshielding change due to the variation of the local currents on the hydrogen bonded proton; (b) a reduction in shielding from currents localized on the oxygen atom of the proton donor; (c) a deshielding contribution from currents induced on the oxygen atom of the proton acceptor. Except for the water dimer, contributions (a), (b) and (c) are of comparable importance for changes in isotropic shielding. For (H₂O)₂ contributions (a) and (c) are somewhat more important than contribution (b). Contribution (c) is almost totally responsible for the changes in the anistropies of the shielding tensors associated with the hydrogen bonded protons. The proton shielding anisotropy changes which occur on hydrogen bond formation are generally much larger than the corresponding variations in the isotropic values of the shielding tensors. This suggests that proton magnetic shielding anisotropies may be more sensitive measures of features of hydrogen bonding than are isotropic proton shielding constants.