Correlation between the out-of-plane components of magnetizability and central magnetic shielding in unsaturated cyclic molecules

A simple classical model of magnetic-field induced π-electron flow is discussed, showing that the contribution to the σ(∥) out-of-plane component of the virtual magnetic shielding provided by π-ring currents, at points P along the C(n) axis of cyclic planar unsaturated hydrocarbons C(n)H(n) with D(nh) symmetry, in the presence of a magnetic field B(ext) at right angles to the σ(h) plane, is, with good approximation, connected with the π-electron contribution to the out-of-plane component of the magnetizability, ξ(∥). The relationship is σ(∥)(h) = -(μ(0)/2π)(s(2) + h(2))(-3/2)ξ(∥), where s is the distance of a C nucleus from the center of the carbon ring, and h is the distance of P from σ(h). The ring current susceptibility, that is, the strength of the π currents, expressed in nA/T (nano ampe?re per tesla) within the SI system of units, is given by ?I/?B(ext) = -ξ(∥)/(πs(2)), which can be used as a reliable virtual measure of magnetotropicity and relative π-electron mobility in isoelectronic systems. Criteria for the practicality of the proposed ring current model are discussed.     

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) .     

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.     

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.     

Diatropicity of tetraazanaphthalenes

Tetraazanaphthalenes are diatropic molecules, whose magnetic response to a magnetic field perpendicular to the molecular plane closely resembles that of naphthalene. The out-of-plane component of the magnetic susceptibility tensor and its strong anisotropy can be used as quantifiers of magnetic aromaticity. Maps showing streamlines and modulus of the current density field provide clear evidence for diatropicity of these systems. They also explain the strong anisotropy of carbon and nitrogen magnetic shielding, which is determined by the big out-of-plane component of the nuclear shielding tensor. The electronic ring currents observed in the map deshield the nuclei of ring hydrogens by enforcing the local magnetic field and diminishing the out-of-plane component of proton shielding.     

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.     

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.     

NMR shielding constants in PH3, absolute shielding scale, and the nuclear magnetic moment of 31P

Ab initio values of the absolute shielding constants of phosphorus and hydrogen in PH(3) were determined, and their accuracy is discussed. In particular, we analyzed the relativistic corrections to nuclear magnetic resonance (NMR) shielding constants, comparing the constants computed using the four-component Dirac-Hartree-Fock approach, the four-component density functional theory (DFT), and the Breit-Pauli perturbation theory (BPPT) with nonrelativistic Hartree-Fock or DFT reference functions. For the equilibrium geometry, we obtained σ(P) = 624.309 ppm and σ(H) = 29.761 ppm. Resonance frequencies of both nuclei were measured in gas-phase NMR experiments, and the results were extrapolated to zero density to provide the frequency ratio for an isolated PH(3) molecule. This ratio, together with the computed shielding constants, was used to determine a new value of the nuclear magnetic dipole moment of (31)P: μ(P) = 1.1309246(50) μ(N).     

Ring current effects on nuclear magnetic shielding of carbon in the benzene molecule

The differential Biot-Savart law of classical electrodynamics was applied to develop a ring current model for the magnetic shielding of the carbon nucleus in benzene. It is shown that the local effect of the pi currents, induced by a magnetic field normal to the molecular plane, on the sigmaC out-of-plane shielding tensor component vanishes. However, approximately 10% of sigmaC is due to the shielding contributions from pi current density in the region of the other carbon atoms. Magnetic shielding density maps obtained via quantum mechanical procedures confirm the predictions of the classical model. Copyright (c) 2004 John Wiley & Sons, Ltd.     

Aromaticity of N-heterocyclic carbene and its analogues: Magnetically induced ring current perspective

The N-heterocyclic carbene, imidazole-2-ylidene, and its main group (13-15) analogues contain cyclically conjugated 6π electrons. Experimental ¹H nuclear magnetic resonance (NMR) spectra suggest an increase in aromaticity along a period from left to right. Whereas the order along a group is as follows: period 2 > period 5 > period 4 > period 3 due to change in structure. To understand the order of aromaticity, the magnetically induced ring currents of the molecules are calculated using aromatic ring current shielding, gauge-including magnetically induced currents (GIMIC) method and Stanger's σ-model applying the gauge-including atomic orbitals NMR technique. It is found that GIMIC best describes the order of aromaticity especially along a group where current-profile changes on the bivalent atom down a group due to change in electron density. Moreover, the GIMIC provides the visualization of current by sign modulus and the anisotropy of the induced current density plots.     

Interatomic magnetizability: a QTAIM-based approach toward deciphering magnetic aromaticity

Interatomic magnetizability provides insight into the extent of electronic current density between two adjacent atomic basins. By studying a number of well-known aromatic, nonaromatic, and antiaromatic molecules, it is demonstrated that interatomic magnetizability (bond magnetizability) not only is able to verify the exact nature of aromaticity/antiaromaticity among different molecules, but also can distinguish the correct aromaticity order among sets of aromatic/antiaromatic molecules. The interatomic magnetizability is a direct measure of the current flux between two adjacent atomic basins and is the first QTAIM-derived index that evaluates aromaticity based on a response property, that is, magnetizability. Bond magnetizability is easy to compute, straightforward to interpret, and can be employed to evaluate the pure π- or σ-orbital contributions to magnetic aromaticity.     

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]     

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.     

Aromaticity of ring carbo-mers of [N]annulenes and [N]cycloalkanes

Maps of current density induced by a perpendicular external magnetic field are calculated at the ipsocentric CTOCD-DZ/6-31G**//B3PW91/6-31G** level for ring carbo-mers of [N]-annulenes (closed-shell singlet states of C(3N)H, N = 3 to 7, with q = -1, 0, +1, 0, -1, respectively, and also the triplet ground state for N = 4) and of [N]-cycloalkanes (C(3N)H(qN), N = 3, 4, 5). Strong four-electron diatropic ring currents indicate conventional pi aromaticity for all the singlet and triplet carbo-[N]annulenes studied, with the exception of C(12)H(4), where instead the strong two-electron paratropic ring current is the signature of pi antiaromaticity. The carbo-[N]cycloalkanes (also known as [N]pericyclynes) show only localized pi currents, consistent with non-aromaticity. There is no indication of a 'homo-aromatic' ring current attributable to the in-plane pi orbitals of the inserted C2 units in any of the maps. Consequences for the interpretation of ELF (electron localisation function) populations are discussed.     

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.     

Calculations of the contribution of ring currents to the chemical shielding anisotropy

Ring currents can exert a large effect upon the chemical shielding of NMR resonances. The analytical expression developed by Waugh and Fessenden (Waugh, J. S.; Fessenden, R. W. J. Am. Chem. Soc. 1957, 79, 846) and Johnson and Bovey (Johnson, C. E.; Bovey, F. A. J. Chem. Phys. 1957, 29, 1012) only quantifies the contribution of ring currents to the isotropic component of the shielding tensor. In the work described here an additional analytical expression is developed so that the contribution of ring currents to the full shielding tensor can be calculated, allowing an estimate of the influence of ring currents upon the chemical shielding anisotropy (CSA, Deltasigma). To test that this pair of analytical expressions can provide a reasonable estimate of the contribution of ring currents to the full shielding tensor a series of density functional calculations (DFT) were carried out. A shielding tensor in a model compound was calculated in two distinct ways. For the first series, DFT shielding calculations of the model compound were carried out in the presence of a benzene ring. For the second series a ring current contribution to the shielding tensor was calculated via the new expressions, and this was added to the result of a DFT shielding calculation which used in place of benzene the nonaromatic analogue 1,3 cyclohexadiene. The two series of results proved to be in excellent agreement. The pair of analytical expressions are used to calculate ring current contributions to the CSA (Deltasigma) of 1H(N) backbone amide resonances in a structure of the second type 2 module from the protein fibronectin. Significant CSA variations are predicted in particular for the 1H(N) of G42 which is most likely involved in a N-H...tpi aromatic hydrogen bond.     

Magnetic shielding tensor components in a highly strained phosphorus molecule: 1,2,3-triphenylphosphirene

Using the magic-angle spinning (MAS) technique the principal values of the ³¹P NMR magnetic shielding tensor were measured in 1,2,3-triphenylphosphirene: σ₁₁ = −58; σ₂₂ = 23 and σ₃₃ = 636 ppm. The anisotropy is quite large for a three-coordinate phosphorus compound. The high-field component σ₃₃ corresponds to the direction perpendicular to the ring, and is responsible for the high-field resonance in the isotropie phase.     

Evaluating Shielding‐Based Ring‐Current Models by Using the Gauge‐Including Magnetically Induced Current Method

Magnetically induced current densities and integrated ring‐current strength susceptibilities have been calculated at the density functional theory (DFT) level for a test set consisting of 17 ring‐shaped molecules using the gauge‐including magnetically induced current (GIMIC) method. Reliable values for the ring‐current strengths have been obtained by performing numerical integration of the current‐density susceptibility passing a cut plane perpendicularly to the molecular ring. The current densities and ring current strengths were calculated at the DFT level using the B3LYP functional and def2‐TZVP basis sets. Current densities and ring‐current strengths have also been calculated at the Hartree‐Fock self‐consistent field (HF‐SCF) level using Dunning’s aug‐cc‐pVTZ basis sets, which allow a direct comparison with ring‐current strengths that have previously been estimated using ring‐current models based on magnetic shielding calculations. Current density calculations at both levels of theory show that the magnetic shielding based ring‐current models are not a very accurate means to estimate the magnetically induced ring current strengths, whereas they provide qualitatively the correct aromaticity trends for the studied molecules.     

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.     

Ring-current models from the differential Biot-Savart law

The differential Biot-Savart law provides simple models for the pi ring currents induced in diatropic and paratropic planar conjugated molecules by a perpendicular magnetic field. The model predictions are confirmed by ab initio maps of nuclear magnetic shielding density. The effects on the protons and on the ring carbon atoms from the closest and furthest segments of the current loop are easily interpreted. [structure: see text]