Ring-Current Models from the Differential Biot-Savart Law

Aromaticity in the Electronic Ground and Lowest Triplet States of Molecules with Fused Thiophene Rings

Analysis of the variations of the off-nucleus isotropic magnetic shielding, σiso(r), around thiophene, thienothiophenes, dithienothiophenes and sulflowers in their electronic ground (S0) and lowest triplet (T1) states reveals that some of the features of aromaticity and bonding in these molecules do not fit in with predictions based on the popular Hückel's and Baird's rules. Despite having 4n π electrons, the S0 states of the sulflowers are shown to be aromatic, due to the local aromaticities of the individual thiophene rings. To reduce its T1 antiaromaticity, the geometry of thiophene changes considerably between S0 and T1: In addition to losing planarity, the carbon-carbon two 'double' and one 'single' bonds in S0 turn into two 'single' and one 'double' bonds in T1. Well-defined Baird-style aromaticity reversals are observed between the S0 and T1 states of only three of the twelve thiophene-based compounds investigated in this work, in contrast, the sulflower with six thiophene rings which is weakly aromatic in S0 becomes more aromatic in T1. The results suggest that the change in aromaticity between the S0 and T1 states in longer chains of fused rings is likely to affect mostly the central ring (or the pair of central rings); rings sufficiently far away from the central ring(s) can retain aromatic character.

Revisiting Gauge-Independent Kinetic Energy Densities in Meta-GGAs and Local Hybrid Calculations of Magnetizabilities

In a recent study [J. Chem. Theory Comput. 2021, 17, 1457-1468], some of us examined the accuracy of magnetizabilities calculated with density functionals representing the local density approximation (LDA), generalized gradient approximation (GGA), meta-GGA (mGGA), as well as global hybrid (GH) and range-separated (RS) hybrid functionals by assessment against accurate reference values obtained with coupled-cluster theory with singles, doubles, and perturbative triples [CCSD(T)]. Our study was later extended to local hybrid (LH) functionals by Holzer et al. [J. Chem. Theory Comput. 2021, 17, 2928-2947]; in this work, we examine a larger selection of LH functionals, also including range-separated LH (RSLH) functionals and strong-correlation LH (scLH) functionals. Holzer et al. also studied the importance of the physically correct handling of the magnetic gauge dependence of the kinetic energy density (τ) in mGGA calculations by comparing the Maximoff-Scuseria formulation of τ used in our aforementioned study to the more physical current-density extension derived by Dobson. In this work, we also revisit this comparison with a larger selection of mGGA functionals. We find that the newly tested LH, RSLH, and scLH functionals outperform all of the functionals considered in the previous studies. The various LH functionals afford the seven lowest mean absolute errors while also showing remarkably small standard deviations and mean errors. Most strikingly, the best two functionals are scLHs that also perform remarkably well in cases with significant multiconfigurational character, such as the ozone molecule, which is traditionally excluded from statistical error evaluations due to its large errors with common density functionals.

Current density and molecular magnetic properties

We give an overview of the molecular response to an external magnetic field perturbing quantum mechanical systems. We present state-of-the-art methods for calculating magnetically-induced current-density susceptibilities. We discuss the essence and properties of current-density susceptibilities and how molecular magnetic properties can be calculated from them. We also review the theory of spin-current densities, how relativity affects current densities and magnetic properties. An overview of the magnetic ring-current criterion for aromaticity is given, which has implications on theoretical and experimental research. The recently reported theory of antiaromaticity and how molecular symmetry affects the magnetic response are discussed and applied to closed-shell paramagnetic molecules. The topology of magnetically induced current densities and its consequences for molecular magnetic properties are also presented with twisted and toroidal molecules as examples.

Spatial Contributions to 1H NMR Chemical Shifts of Free-Base Porphyrinoids

A recently developed methodology for calculating, analyzing, and visualizing nuclear magnetic shielding densities is used for studying spatial contributions including ring-current contributions to 1H nuclear magnetic resonance (NMR) chemical shifts of aromatic and anti-aromatic free-base porphyrinoids. Our approach allows a visual inspection of the spatial origin of the positive (shielding) and negative (deshielding) contributions to the nuclear magnetic shielding constants. Diatropic and paratropic current-density fluxes yield both shielding and deshielding contributions implying that not merely the tropicity of the current density determines whether the contribution has a shielding or deshielding character. Instead the shielding or deshielding contribution is determined by the direction of the current-density flux with respect to the studied nucleus.

Magnetic Shielding Study of Bonding and Aromaticity in Corannulene and Coronene

Bonding and aromaticity in the bowl-shaped C5v and planar D5h geometries of corannulene and the planar D6h geometry of coronene are investigated using 3D isosurfaces and 2D contour plots of the isotropic magnetic shielding σiso(r) and, for planar geometries, of the out-of-plane component of the shielding tensor σzz(r). Corannulene and coronene both feature conjoined shielded “doughnuts” around a peripheral six-membered carbon ring, suggesting strong bonding interactions and aromatic stability; a deshielded region inside the hub ring of corannulene indicates that this ring is antiaromatic, more so in planar corannulene. The switch from the planar to the bowl-shaped geometry of corannulene is shown to enhance both bonding and the local aromaticities of the five- and six-membered rings; these factors, in addition to ring strain reduction, favour the bowl-shaped geometry. The most and least shielded bonds in both corannulene and coronene turn out to be the spoke and hub bonds, respectively. The higher π electron activity over spoke bonds in planar corannulene and coronene is supported by σzz(r) contour plots in planes 1 Å above the respective molecular planes; these findings about spoke bonds are somewhat unexpected, given that ring current studies indicate next to no currents over spoke bonds.

Benchmarking Magnetizabilities with Recent Density Functionals

We have assessed the accuracy of the magnetic properties of a set of 51 density functional approximations, including both recently published and already established functionals. The accuracy assessment considers a series of 27 small molecules and is based on comparing the predicted magnetizabilities to literature reference values calculated using coupled-cluster theory with full singles and doubles and perturbative triples [CCSD(T)] employing large basis sets. The most accurate magnetizabilities, defined as the smallest mean absolute error, are obtained with the BHandHLYP functional. Three of the six studied Berkeley functionals and the three range-separated Florida functionals also yield accurate magnetizabilities. Also, some older functionals like CAM-B3LYP, KT1, BHLYP (BHandH), B3LYP, and PBE0 perform rather well. In contrast, unsatisfactory performance is generally obtained with Minnesota functionals, which are therefore not recommended for calculations of magnetically induced current density susceptibilities and related magnetic properties such as magnetizabilities and nuclear magnetic shieldings. We also demonstrate that magnetizabilities can be calculated by numerical integration of magnetizability density; we have implemented this approach as a new feature in the gauge-including magnetically induced current (GIMIC) method. Magnetizabilities can be calculated from magnetically induced current density susceptibilities within this approach even when analytical approaches for magnetizabilities as the second derivative of the energy have not been implemented. The magnetizability density can also be visualized, providing additional information that is not otherwise easily accessible on the spatial origin of magnetizabilities.

Spatial Contributions to Nuclear Magnetic Shieldings

We develop a methodology for calculating, analyzing, and visualizing nuclear magnetic shielding densities which are calculated from the current density via the Biot-Savart relation. Atomic contributions to nuclear magnetic shielding constants can be estimated within our framework with a Becke partitioning scheme. The new features have been implemented in the GIMIC program and are applied in this work to the study of the 1H and 13C nuclear magnetic shieldings in benzene (C6H6) and cyclobutadiene (C4H4). The new methodology allows a visual inspection of the spatial origins of the positive (shielding) and negative (deshielding) contributions to the nuclear magnetic shielding constant of a single nucleus, something which has not been hitherto easily accomplished. Analysis of the shielding densities shows that diatropic and paratropic current-density fluxes yield both shielding and deshielding contributions, as the shielding or deshielding is determined by the direction of the current-density flux with respect to the studied nucleus instead of the tropicity. Becke partitioning of the magnetic shieldings shows that the magnetic shielding contributions mainly originate from the studied atom and its nearest neighbors, confirming the localized character of nuclear magnetic shieldings.

Anisotropy of the vorticity tensor as a magnetic indicator of aromaticity

The vorticity vector of the current density J^B, induced in the electron cloud of a molecule by a magnetic field B, is defined by V^B = ∇ × J^B.

Frequency-dependent current density tensors as density functions of dynamic polarizabilities

Relationships accounting for contributions to the first-order charge density ρ⁽¹⁾ and current density J⁽¹⁾, induced in the electrons of a molecule by a monochromatic plane wave, have been obtained via time-dependent quantum mechanical perturbation theory. Their gauge invariance and invariance in passive translations of the coordinate system have been demonstrated (i) within the (long wavelengths) electric dipole approximation, in which only the electric field contributions to these densities are required, and (ii) within the electric quadrupole approximation, in which nonseparable terms provided by the magnetic field and by the electric field gradient, assumed uniform over the molecular domain, are needed. It is shown that the physical meaning of current density tensors depending on the frequency ω of the monochromatic wave shone on the molecule, and corresponding to derivatives of the current density J⁽¹⁾(r, ω) with respect to components of the perturbing fields, is that of property density. Therefore, frequency-dependent current density tensors can be interpreted as the integrand function in 3d-space integrals defining dynamic molecular response tensors, e.g., electric dipole, mixed electric dipole-magnetic dipole, and electric dipole-quadrupole polarizabilites. Plots of current density tensors are expected to provide important information on the molecular domains giving dominant contributions to these properties.

Cyclic tris-[5]helicenes with single and triple twisted Möbius topologies and Möbius aromaticity

A number of singly (180°) twisted, largely single-stranded and thus conformationally rather fragile, Möbius molecules have been synthesized within the last 15 years, which are aromatic with 4n electrons, thus violating the Hückel rule. Annulenes with significantly higher twist (e.g. 540°) that retain a full cyclic conjugation path have been elusive, mainly because of the high strain and loss of orbital overlap. Recently, a topological strategy was devised to project the "twist" into "writhe", thus reducing the strain. However, orbital overlap was still severely reduced within the flexible building blocks. We now present a single and a triple twisted annulene with fully conjugated peripheries. They are unique in their pronounced band shape and conformational robustness as they are made up of three fully kata-condensed [5]helicene fragments. The triple twisted molecule exhibits a strong diatropic ring current in the outer periphery, even though the π system includes 4n electrons. The diatropic current is counterbalanced by a paratropic current in the σ system, resulting in no net manifestation of macrocyclic aromaticity. The key step of the synthesis of both Möbius compounds is a Perkin condensation of complementary bifunctional bismaleates leading to a flexible macrocycle containing alternating benzene and biphenyl fragments. Subsequent photocyclization yields a separable mixture of rigid diastereomeric tris-helicene macrocycles of the above topologies.

Strict Correlation of HOMO Topology and Magnetic Aromaticity Indices in d-Block Metalloaromatics

Magnetic aromaticity and antiaromaticity of closed shell metalloaromatics with 4d transition metals (Nb, Tc, Rh) is strictly correlated with the orbital topology (Möbius or Hückel) of their π-HOMO, investigated computationally with DFT methods. A surprisingly simple rule emerged: the metallacycle is aromatic (antiaromatic) when the number of π MOs is even and the π-HOMO is of Möbius (Hückel) topology-and vice versa when the number of π MOs is odd.

Topological definition of ring currents

A definition of ring currents in a velocity vector field is proposed according to topological criteria: ring currents are axial vortices confined in, or rotating beyond, a separatrix, i.e., the boundary which marks the limits of the vortex.

Magnetic criteria of aromaticity

This review describes the current state of magnetic criteria of aromaticity. The introduction contains the fundamentals of ring currents in aromatic and antiaromatic systems, followed by a brief description of experimental and computational tools: NMR, diamagnetic susceptibility exaltation, current density analyses (CDA) and nucleus independent chemical shifts (NICS). This is followed by more comprehensive chapters: NMR - focusing on the work of R. Mitchell - NICS and CDA - describing the progress and development of the methods to their current state and presenting some examples of representative work.

Polygonal current model: an effective quantifier of aromaticity on the magnetic criterion

To explain peculiar effects of electron delocalization on the magnetic response of planar cyclic molecules, a basic model that accounts for their actual geometrical structure has been developed by integrating the differential Biot-Savart law. Such a model, based on a single polygonal circuit with ideal features, is shown to be applicable to electrically neutral or charged monocyclic compounds, as well as linear polycyclic condensed hydrocarbons. Two theoretical quantities, easily computed via quantum chemistry codes (the out-of-plane components of the magnetizability, ξ∥, and the magnetic shielding σ∥(h) of points P on the symmetry axis orthogonal to the molecular plane, at distance h from the center of mass) are shown to be linearly connected, for example, for monocyclic structures, via the relationship σ∥(h) = ±(μ0/2π)ξ∥D(h), where D(h) is a simple function of geometrical parameters. Equations of this type are useful to rationalize scan profiles of magnetic shielding and nucleus-independent chemical shift along the highest symmetry axis. For a regular polygon, D(h) depends approximately on the third inverse power of the distance d of the vertices from the center, and ξ∥ is proportional to the area of the polygon, that is, ∼d(2); hence, the shielding σ∥(0) and the related nucleus-independent chemical shift NICS∥(0) are unsafe quantifiers of magnetotropicity; they are biased by a spurious geometrical dependence on d(-1), incorrectly exhalting them in cyclic systems with smaller size. A more reliable magnetotropicity measure for a cyclic compound, in the presence of a magnetic field Bext applied at right angles to the molecular plane, is defined within the polygonal current model by the current susceptibility or current strength, ∂I/∂Bext = -ξ∥/Aeff, expressed in nanoampère per tesla, where Aeff is a properly defined area enclosed with the polygonal circuit. An extended numerical test on a wide series of mono- and polycyclic compounds and a comparison with corresponding ab initio current susceptibilities prove the superior quality of this indicator over other commonly employed aromaticity/antiaromaticity benchmarks on the magnetic criterion.

Obtaining relative induced ring currents quantitatively from NICS

A model for obtaining the sigma-only effect on the nucleus-independent chemical shift (NICS) is introduced. By subtracting the values obtained for the "sigma-only" model from the respective values of the conjugated system it is possible to obtain the NICS values that originate from the pi system only. This procedure allows obtaining a quantitative measure of the relative intensities of diatropic and paratropic ring currents in different systems with a built-in measure of their accuracy. The scope and limitations of the model are discussed.

Spatial ring current model for the prismane molecule

Spatial models of magnetic-field induced electronic ring currents have been constructed for the prismane molecule via stagnation graphs and current density maps. These tools provide an insight into the complicated phenomenology resulting from competition of diatropic and paratropic regimes that determine the magnitude of various components of magnetic susceptibility and magnetic shielding of hydrogen and carbon nuclei. Shielding density maps show that the differential Biot-Savart law, along with an atlas of the current density field, explains magnetic shielding at hydrogen and carbon nuclei and virtual shielding at ring and cage centers.

Assessment of σ-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.

Spatial Ring Current Model of the [2.2]Paracyclophane Molecule

A representation of the current density induced in the [2.2]paracyclophane molecule by a homogeneous magnetic field parallel to the line joining the centers of the phenylene rings is given in compact form by a stagnation graph that conveys essential information. Analogous graphs were obtained for two perpendicular directions. Plots of streamlines are also reported to complete a ring current model that has been proved useful to understand the magnetropicity of the system. Stagnation graphs, maps of streamlines and moduli of the current density, and plots of Biot-Savart magnetic shielding density provide a basic tool kit for rationalizing magnetic response of complex systems.

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.

Which NICS aromaticity index for planar pi rings is best?

Five increasingly sophisticated aromaticity indexes, based on nucleus-independent chemical shifts (NICS), were evaluated against a uniform set of aromatic stabilization energies (ASE) for 75 mono- and polyheterocyclic five-membered rings. While acceptable statistical correlations were given by all of the NICS methods, the most fundamentally grounded index, NICS(0)pizz (based on the pi contribution to the out-of-plane zz tensor component), performed best statistically (cc=0.980) and in practice. The easily computable NICS(1)zz index is a useful alternative (cc=0.968).

σ and π contributions to the induced magnetic field: Indicators for the mobility of electrons in molecules

The authors discuss the role of the sigma and pi contributions to the induced magnetic field for simple hydrocarbons containing a double or a triple bond, as well as for benzene and cyclobutadiene. While the magnetic field induced by the sigma electrons is short-ranged, the pi system is responsible for the formation of long-range cones. These cones influence the chemical shift of atoms by additional shielding (for aromatic) or deshielding (for antiaromatic molecules) contributions. While the hydrogen atoms of benzene are found to lie within the deshielded region of the magnetic field induced by the pi electrons, they are shielded by the total induced magnetic field. The induced magnetic field of the pi electrons support Pople's model on the basis of first-principles calculations.

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]

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.

Downfield Proton Chemical Shifts Are Not Reliable Aromaticity Indicators

[structure: see text] The downfield chemical shifts of arene hydrogens (delta (1)H) are due only in part to the pi ring current contribution [sigma(//)(pi)]; local framework effects are equally important. Neither proton chemical shifts nor even sigma(||)(pi) tensor elements, per se, are reliable aromaticity indicators. Unsaturated polycyclic hydrocarbons with nonaromatic quinoid structures have delta (1)H and sigma(//)(pi) values in the "aromatic range". Conversely, numerous aromatic protons, including those in five-membered ring heterocycles, resonate in the "nonaromatic range".