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.     

The ring current model of the pentaprismane molecule

Three-dimensional models of the quantum-mechanical current density J(B) , induced in the electron cloud of the C(10)H(10) pentaprismane molecule by a magnetic field B applied along the C(5) (a C(2)) symmetry axis, orthogonal to the pentagonal (a rectangular) face, and denoted by B(‖) (B(⊥)), have been constructed. Predictions of near Hartree-Fock quality are reported for the diagonal components of magnetic tensors, magnetizability (ξ), nuclear shielding of carbon (σ(C)) and hydrogen (σ(H)), and virtual shielding at the center of mass (σ(CM)). The complicated spatial features of the induced electronic current-density field have been rationalized and compactly described via stagnation graphs that elucidate the details of its topological structure. A representation of J(B) is obtained by three-dimensional perspective plots and by planar maps visualizing phase portraits of electron flow in a series of molecular domains. Both streamline J(B) /|J(B) | and modulus |J(B) | are analyzed. These graphic tools illustrate the competition between diatropic and paratropic regimes which determine the magnitude of various components of magnetizability and magnetic shielding of hydrogen and carbon nuclei. Shielding density maps show that the differential Biot-Savart law explains magnetic shielding at hydrogen and carbon nuclei, and virtual shielding at ring and cage centers. Similarities and/or contrasting ring current effects on magnetotropicity are discussed by a comparison with triprismane C(6)H(6) and cubane C(8)H(8) .     

Calculation of magnetic properties of HF,H2O,NH3, and CH4 molecules using a longitudinal gauge for the vector potential

A transformation of the transverse Coulomb vector potential was implemented to calculate molecular magnetic properties via the random-phase approximation (RPA) within the framework of a “longitudinal gauge.” In this gauge, the diamagnetic contribution to magnetic susceptibility is a tensor with equal diagonal components as in atoms, irrespective of molecular symmetry, whereas diagonal and average diamagnetic contributions to the nuclear magnetic shielding are the same as in the Coulomb gauge. Near-Hartree–Fock magnetic susceptibility and nuclear magnetic shielding tensors were evaluated for a set of small molecules, HF, H₂O, NH₃, and CH₄, employing extended Gaussian basis sets. The peculiar features of the longitudinal gauge, and the fulfillment of a series of sum rules involving the virial operator, which must be satisfied to guarantee gauge invariance of total magnetic tensors, were exploited to check the degree of convergence of theoretical values and to estimate the corresponding Hartree–Fock limit for the properties. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 31–45, 1998     

Assessment of sigma-diatropicity of the cyclopropane molecule

Spatial models of the current density field induced in the cyclopropane molecule by stationary, homogeneous magnetic fields, parallel to either the C3 or the C2 symmetry axis, have been constructed. A compact, abridged representation of the models is given via stagnation graphs that convey essential information. Maps of streamlines and moduli are also reported to complete current models that have proven useful to rationalize magnetic tensor properties, that is, magnetizability, 1H and 13C nuclear shieldings, and magnetic shielding along the C3 symmetry axis. Plots of Biot-Savart magnetic shielding density combined with current density visualization yield an accurate, detailed account of the shielding mechanisms. The magnetropicity of the system described by the current density model is fully consistent with the magnitude of magnetic tensors calculated at near Hartree-Fock level. In a field perpendicular to the molecular plane, cyclopropane sustains a diatropic sigma-ring current with the following peculiar features: (i) it follows the molecular periphery rather than the CC framework; (ii) it bifurcates in the proximity of the methylene moieties flowing along the CH bonds, both above and below the sigma(h) plane; (iii) it has an effect on the values of response properties, although it is not as large as expected from naive arguments (e.g., the center-of-mass value of the magnetic shielding constant is dominated by in-plane components rather than the out-of-plane component, which is in contrast to pi-aromatic systems such as benzene); (iv) it has a negligible effect on the strong anisotropy of carbon magnetic shielding, which is shown to arise from local currents. No evidence for strong diatropism, and therefore sigma-aromaticity of the cyclopropane molecule, was found on the magnetic criterion.     

Beyond NICS: estimation of the magnetotropicity of inorganic unsaturated planar rings

A simple classical model of magnetic-field induced electron flow is used to evaluate the ring current strength for a few inorganic monocyclic compounds: B(3)H(3)N(3), B(3)H(3)O(3), P(6), N(6), Si(6)H(6), N, Al and H(6). It is shown that, for these neutral and charged systems, sustaining delocalized electron currents in the presence of a magnetic field B(ext) orthogonal to the σ(h) plane, the out-of-plane component of the nuclear magnetic shielding along the central axis is connected to the out-of-plane magnetizability by a simple equation, involving the radius of an average loop of current. A novel estimate of this effective radius is provided. Reliable ring current susceptibilities (that is, current strengths) can be evaluated by a simple relationship, using the out-of-plane components of nuclear shielding and magnetizability tensors. The accuracy of the current susceptibilities calculated by the classical model is established by comparison with corresponding ab initio estimates obtained by integrating the quantum mechanical current-density vector field. The out-of-plane components of nuclear shielding and magnetizability are both strongly biased by the molecular geometry. Their combined use to estimate the ring current susceptibility offers a quantifier of magnetotropicity more reliable than (i) the ξ(∥) out-of-plane component of magnetizability, (ii) the σ(∥)(CM) out-of-plane component of the magnetic shielding at the center of mass, widely reported as NICS(∥)(0) = -σ(∥)(CM). The inadequacy of these commonly adopted magnetotropicity measures is demonstrated by comparing a set of related molecules, C(6)H(6) and Si(6)H(6), N(6) and P(6).     

Breakdown of the pseudopotential approximation for magnetizabilities and electric multipole moments. II. The importance of gauge invariance for large-core semi-local pseudopotentials

In a previous paper we pointed out that core contributions to the static magnetizability are non-negligible, and can therefore lead to erroneous results within the pseudopotential approximation [P. Schwerdtfeger, B. Assadollahzadeh, U. Rohrmann, R. Scha?fer, and J. R. Cheeseman, J. Chem. Phys. 134, 204102 (2011)]. In a recent paper van Wu?llen showed that additional terms arising from the gauge-invariant condition for the semi-local part of a pseudopotential operator can lead to non-negligible contributions to the magnetizability tensor, which are sensitive to the gauge origin and basis set chosen [C. van Wu?llen, J. Chem. Phys. 136, 114110 (2012)]. These terms were neglected in previous calculations as they were assumed to be small. In this paper we analyze the importance of the gauge-dependent semi-local pseudopotential correction term in detail for AuF and clusters of Sn showing that it leads indeed to very large corrections to the paramagnetic term for large-core pseudopotentials. Without this correction the results become very sensitive to the basis set applied. This now resolves some of the unusual large paramagnetic contributions reported before for both AuF and Sn(2).     

Understanding the ring current effects on magnetic shielding of hydrogen and carbon nuclei in naphthalene and anthracene

The local response to an external magnetic field normal to the molecular plane of naphthalene and anthracene was investigated via current density and magnetic shielding density maps. The Biot-Savart law shows that the deshielding caused by pi-ring currents in naphthalene is stronger for alpha- than for beta-protons due to geometrical factors. The shielding tensor of the carbon nuclei in both molecules is strongly anisotropic and its out-of-plane component determines the up-field chemical shift of (13)C in nuclear magnetic resonance spectra. The pi-ring currents flowing beyond the C-skeleton in front of a probe carbon nucleus, and on remote parts of the molecular perimeter, yield positive contributions to the out-of-plane component of carbon shielding as big as approximately 10-15% of the total values. Near Hartree-Fock estimates of magnetizability and magnetic shielding at the nuclei fully consistent with the current model are reported.     

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.     

The nuclear magnetic resonance line shapes of Xe in the cages of clathrate hydrates

We report, for the first time, a prediction of the line shapes that would be observed in the (129)Xe nuclear magnetic resonance (NMR) spectrum of xenon in the cages of clathrate hydrates. We use the dimer tensor model to represent pairwise contributions to the intermolecular magnetic shielding tensor for Xe at a specific location in a clathrate cage. The individual tensor components from quantum mechanical calculations in clathrate hydrate structure I are represented by contributions from parallel and perpendicular tensor components of Xe-O and Xe-H dimers. Subsequently these dimer tensor components are used to reconstruct the full magnetic shielding tensor for Xe at an arbitrary location in a clathrate cage. The reconstructed tensors are employed in canonical Monte Carlo simulations to find the Xe shielding tensor component along a particular magnetic field direction. The shielding tensor component weighted according to the probability of finding a crystal fragment oriented along this direction in a polycrystalline sample leads to a predicted line shape. Using the same set of Xe-O and Xe-H shielding functions and the same Xe-O and Xe-H potential functions we calculate the Xe NMR spectra of Xe atom in 12 distinct cage types in clathrate hydrates structures I, II, H, and bromine hydrate. Agreement with experimental spectra in terms of the number of unique tensor components and their relative magnitudes is excellent. Agreement with absolute magnitudes of chemical shifts relative to free Xe atom is very good. We predict the Xe line shapes in two cages in which Xe has not yet been observed.     

Molecular magnetic properties via formal annihilation of paramagnetic contribution to electronic current density

A novel procedure for calculating magnetic susceptibilities and nuclear magnetic shieldings in molecules is outlined, based on formal annihilation of transverse paramagnetic contribution to quantum mechanical current density induced within the electron cloud by an external homogeneous, static magnetic field. Within this method all the components of nuclear magnetic shielding are independent of a gauge translation, in any calculation relying on the algebraic approximation, irrespective of size and quality of the gaugeless basis set adopted; magnetic susceptibilities are invariant for center-symmetric molecules (virtual invariance is actually observed for molecules of arbitrary symmetry). Large basis set calculations of near-Hartree-Fock magnetic properties and maps describing a current density field of acetylene molecule carried out via the new procedure are compared with corresponding ones adopting the common origin-coupled Hartree-Fock approach and methods formally annihilating the diamagnetic contribution to the current density. © 1996 John Wiley & Sons, Inc.     

Connection between the nuclear electric shielding tensor and the infrared intensity

The integrated IR intensity can be directly related to the electric shielding tensor of the nuclei in a molecule. This allows us to measure, by IR spectroscopy, the effective electric field at the nuclei of a molecule immersed in an external electric field. Relations exist with the dynamic polarizability and the magnetizability, which implies the possibility of a unified treatment of electric and magnetic second-order properties.     

Why downfield proton chemical shifts are not reliable aromaticity indicators

Traces of magnetizability, traces of magnetic shielding at the hydrogen nuclei, and nucleus-independent chemical shift are not reliable aromaticity quantifiers for planar conjugated hydrocarbons. A measure of aromaticity is provided by the out-of-plane tensor components, whose magnitude is influenced by the pi-ring currents. The failure of nucleus-independent chemical shift in this regard was proved for the molecule shown in the abstract graphic, sustaining a diatropic pi-current. The validity of the ring-current model is reaffirmed. [structure: see text]     

Methods of continuous translation of the origin of the current density revisited

Approaches to the calculation of magnetizability and nuclear magnetic shieldings in a molecule, based on continuous translation of the origin of the magnetic field-induced electronic current density, are reviewed. The connections among apparently unrelated philosophies (Geertsen propagator methods, Keith-Bader continuous set of gauge transformations, and analytical reformulation by Lazzeretti, Malagoli, and Zanasi) are emphasized, and a unitary theoretical scheme is given.     

Orbital Analysis of Carbon‐13 Chemical Shift Tensors Reveals Patterns to Distinguish Fischer and Schrock Carbenes

Fischer and Schrock carbenes display highly deshielded carbon chemical shifts (>250 ppm), in particular Fischer carbenes (>300 ppm). Orbital analysis of the principal components of the chemical shift tensors determined by solid‐state NMR spectroscopy and calculated by a 2‐component DFT method shows specific patterns that act as fingerprints for each type of complex. The calculations highlight the role of the paramagnetic term in the shielding tensor especially in the two most deshielded components (σ₁₁ and σ₂₂). The paramagnetic term of σ₁₁ is dominated by coupling σ(M=C) with π*(M=C) through the angular momentum operator perpendicular to the σ and π M=C bonds. The highly deshielded carbon of Fischer carbenes results from the particularly low‐lying π*(M=C) associated with the CO ligand. A contribution of the coupling of π(M=C) with σ*(M=C) is found for Schrock and Ru‐based carbenes, indicating similarities between them, despite their different electronic configurations (d⁰ vs. d⁶).     

A comparison of two-component and four-component approaches for calculations of spin-spin coupling constants and NMR shielding constants of transition metal cyanides

Relativistic density functional theory (DFT) calculations of nuclear spin-spin coupling constants and shielding constants have been performed for selected transition metal (11th and 12th group of periodic table) and thallium cyanides. The calculations have been carried out using zeroth-order regular approximation (ZORA) Hamiltonian and four-component Dirac-Kohn-Sham (DKS) theory with different nonrelativistic exchange-correlation functionals. Two recent approaches for representing the magnetic balance (MB) between the large and small components of four-component spinors, namely, mDKS-RMB and sMB, have been employed for shielding tensor calculations and their results have been compared. Relativistic effects have also been analysed in terms of scalar and spin-orbit contributions at the two-component level of theory, including discussion of heavy-atom-on-light-atom effects for (1)J(CN), σ(C), and σ(N). The results for molecules containing metals from 4th row of periodic table show that relativistic effects for them are small (especially for spin-spin coupling constants). The biggest effects are observed for the 6th row where nonrelativistic theory reproduces only about 50%-70% of the two-component ZORA results for (1)J(MeC) and about 75% for heavy metal shielding constants. It is important to employ a full Dirac picture for calculations of heavy metal shielding constants, since ZORA reproduces only 75%-90% of the DKS results. Smaller discrepancies between ZORA-DFT and DKS are observed for nuclear spin-spin coupling constants. No significant differences are observed between the results obtained using mDKS-RMB and sMB approaches for magnetic balance in four-component calculations of the shielding constants.     

Ab initio study of the ground state properties of molecular oxygen

The electric and magnetic properties of the ground state of oxygen molecule are calculated by multiconfiguration self-consisted field (MCSCF) method and compared with experimental data: the quadrupole moment, polarizability, the 17O nuclear quadrupole coupling constant, magnetizability tensor, nuclear spin-rotation coupling constant and rotational g factor. The last two constants are calculated for all possible isotope modifications. The rotational, ESR and NMR spectra are discussed. Fermi-contact hyperfine coupling parameter is additionally estimated by different methods. The NMR chemical shielding tensor for 17O16O species at high temperature limit (without electron spin contribution) is predicted. Potential energy curves for 10 excited bound states and the internuclear distance dependence of the studied properties are also presented.     

Formal relations connecting different approaches to calculate relativistic effects on molecular magnetic properties

In the present work a set of formal relations connecting different approaches to calculate relativistic effects on magnetic molecular properties are proven. The linear response (LR) within the elimination of the small component (ESC), Breit Pauli, and minimal-coupling approaches are compared. To this end, the leading order ESC reduction of operators within the minimal-coupling four-component approach is carried out. The equivalence of all three approaches within the ESC approximation is proven. It is numerically verified for the NMR nuclear-magnetic shielding tensor taking HX and CH3X (X=Br,I) as model compounds. Formal relations proving the gauge origin invariance of the full relativistic effect on the NMR nuclear-magnetic shielding tensor within the LR-ESC approach are presented.     

A solid-state 55Mn NMR spectroscopy and DFT investigation of manganese pentacarbonyl compounds

Central transition (55)Mn NMR spectra of several solid manganese pentacarbonyls acquired at magnetic field strengths of 11.75, 17.63, and 21.1 T are presented. The variety of distinct powder sample lineshapes obtained demonstrates the sensitivity of solid-state (55)Mn NMR to the local bonding environment, including the presence of crystallographically unique Mn sites, and facilitates the extraction of the Mn chemical shift anisotropies, CSAs, and the nuclear quadrupolar parameters. The compounds investigated include molecules with approximate C(4v) symmetry, LMn(CO)(5)(L = Cl, Br, I, HgMn(CO)(5), CH(3)) and several molecules of lower symmetry (L = PhCH(2), Ph(3-n)Cl(n)Sn (n= 1, 2, 3)). For these compounds, the Mn CSA values range from <100 ppm for Cl(3)SnMn(CO)(5) to 1260 ppm for ClMn(CO)(5). At 21.1 T the (55)Mn NMR lineshapes are appreciably influenced by the Mn CSA despite the presence of significant (55)Mn quadrupolar coupling constants that range from 8.0 MHz for Cl(3)SnMn(CO)(5) to 35.0 MHz for CH(3)Mn(CO)(5). The breadth of the solid-state (55)Mn NMR spectra of the pentacarbonyl halides is dominated by the CSA at all three applied magnetic fields. DFT calculations of the Mn magnetic shielding tensors reproduce the experimental trends and the magnitude of the CSA is qualitatively rationalized using a molecular orbital, MO, interpretation based on Ramsey's theory of magnetic shielding. In addition to the energy differences between symmetry-appropriate occupied and virtual MOs, the d-character of the Mn MOs is important for determining the paramagnetic shielding contribution to the principal components of the magnetic shielding tensor.     

Relativistic effects on the nuclear magnetic resonance shielding of FX (X = F, Cl, Br, I, and At) molecular systems

We present ab inito full four-component and spin-free calculations of the NMR shielding parameter, σ, in the FX (X = F, Cl, Br, I and At) molecular systems. A different expression that overcomes the traditional non-relativistic (NR) approximation used to calculate the relationship between spin-rotation constants and the paramagnetic terms of σ(p) are given. Large deviations from NR results are obtained for σ(X; X = I and At) and for σ(F; FAt). σ(∥)(p)(I; FI) is zero within the NR approach but -447.4 parts per million from our calculations. The electronic origin of relativistic corrections are analyzed. All passive SO contributions are obtained as a difference between full four-component calculations and spin-free ones. Considering relativistic effects on the anisotropy, we obtain a deviation of 10% for I and 25% for At. σ(∥)(SO)(X) is always negative and σ(∥)(SF)(X) is always positive; the passive SO becomes larger than the SF one for X = Br, I, and At. Both σ(∥)(SO)(X) and σ(⊥)(SO)(X) have a functional dependence such as a Z(X)(b) being the exponent 3.5 and 3.65, respectively. The passive SO contribution to the anisotropy has a similar functional dependence with an exponent of 3.60, meaning that its perpendicular component is larger than its corresponding parallel component.