Physical achirality in geometrically chiral rotamers of hydrazine and boranylborane molecules.

The diagonal components and the trace of tensors which account for chiroptical response of the hydrazine molecule N2 H4 , that is, static anapole magnetizability and frequency-dependent electric dipole-magnetic dipole polarisability, are a function of the ϕ ≡ ∠ H─N─N─H dihedral angle. They vanish for symmetry reasons at ϕ = 0° and ϕ = 180°, corresponding respectively to C2v and C2h point group symmetries, that is, cis and trans conformers characterized by the presence of molecular symmetry planes. Nonetheless, vanishing diagonal components have been observed also in the proximity of ∠ H─N─N─H = 90°, in which the point group symmetry is C2 and hydrazine is unquestionably chiral. In the boranylborane molecule B2 H4 , assuming the B─B bond in the y direction, the ayy component of the anapole magnetizability tensor approximately vanishes for dihedral angles ∠ H─B─B─H corresponding to chiral rotamers which belong to D2 symmetry. Such anomalous effects have been ascribed to physical achirality of these conformers, that is, to their inability to sustain electronic current densities inducing either anapole moments, or electric and magnetic dipole moments, about the chiral axis connecting heavier atoms, as well as perpendicular directions. In other terms, the structure of certain geometrically chiral rotamers may be such that neither toroidal nor helical flow, which determine chiroptical phenomenology, can take place in the presence of perturbing fields parallel or orthogonal to the chiral axis.

Electronic Currents Induced by Optical Fields and Rotatory Power Density in Chiral Molecules.

The electric dipole-magnetic dipole polarizability tensor κ', introduced to interpret the optical activity of chiral molecules, has been expressed in terms of a series of density functions kαß', which can be integrated all over the three-dimensional space to evaluate components καß' and trace καα'. A computational approach to kαß', based on frequency-dependent electronic current densities induced by monochromatic light shining on a probe molecule, has been developed. The dependence of kαß' on the origin of the coordinate system has been investigated in connection with the corresponding change of καß'. It is shown that only the trace kαα' of the density function defined via dynamic current density evaluated using the continuous translation of the origin of the coordinate system is invariant of the origin. Accordingly, this function is recommended as a tool that is quite useful for determining the molecular domains that determine optical activity to a major extent. A series of computations on the hydrogen peroxide molecule, for a number of different HO-OH dihedral angles, is shown to provide a pictorial documentation of the proposed method.

Polygonal current models for polycyclic aromatic hydrocarbons and graphene sheets of various shapes.

Assuming that graphene is an "infinite alternant" polycyclic aromatic hydrocarbon resulting from tessellation of a surface by only six-membered carbon rings, planar fragments of various size and shape (hexagon, triangle, rectangle, and rhombus) have been considered to investigate their response to a magnetic field applied perpendicularly. Allowing for simple polygonal current models, the diatropicity of a series of polycyclic textures has been reliably determined by comparing quantitative indicators, the π-electron contribution to IB , the magnetic field-induced current susceptibility of the peripheral circuit, to ξ∥ and to σ∥(CM)=-NICS∥(CM), respectively the out-of-plane components of the magnetizability tensor and of the magnetic shielding tensor at the center of mass. Extended numerical tests and the analysis based on the polygonal model demonstrate that (i) ξ∥ and σ∥(CM) yield inadequate and sometimes erroneous measures of diatropicity, as they are heavily flawed by spurious geometrical factors, (ii) IB values computed by simple polygonal models are valid quantitative indicators of aromaticity on the magnetic criterion, preferable to others presently available, whenever current susceptibility cannot be calculated ab initio as a flux integral, (iii) the hexagonal shape is the most effective to maximize the strength of π-electron currents over the molecular perimeter, (iv) the edge current strength of triangular and rhombic graphene fragments is usually much smaller than that of hexagonal ones, (v) doping by boron and nitrogen nuclei can regulate and even inhibit peripheral ring currents, (vi) only for very large rectangular fragments can substantial current strengths be expected. © 2017 Wiley Periodicals, Inc.

Computational study of basis set and electron correlation effects on anapole magnetizabilities of chiral molecules.

In the presence of a static, nonhomogeneous magnetic field, represented by the axial vector B at the origin of the coordinate system and by the polar vector C=∇×B, assumed to be spatially uniform, the chiral molecules investigated in this paper carry an orbital electronic anapole, described by the polar vector A. The electronic interaction energy of these molecules in nonordered media is a cross term, coupling B and C via a¯, one third of the trace of the anapole magnetizability aαß tensor, that is, WBC=-a¯B·C. Both A and W(BC) have opposite sign in the two enantiomeric forms, a fact quite remarkable from the conceptual point of view. The magnitude of a¯ predicted in the present computational investigation for five chiral molecules is very small and significantly biased by electron correlation contributions, estimated at the density functional level via three different functionals. © 2016 Wiley Periodicals, Inc.

π-Ring currents in doped coronenes with nitrogen and boron: diatropic-paratropic duality.

The change in the electronic structure of coronene upon doping with nitrogen or boron has been theoretically studied by means of its magnetic properties and magnetic field induced current density maps. The addition of two atoms of nitrogen or boron to the central ring of coronene causes a drastic variation in the delocalization of π-electrons, which does not depend on its nature but instead on its position. Then, doping in the para position makes coronene more aromatic while doping in the meta position makes it to become antiaromatic. The magnetic behavior of the pristine molecule is characterized by two concentric currents flowing in opposite senses that are converted into hemi-perimetric currents in the ortho and meta isomers, so dividing the molecule into aromatic and antiaromatic regions. The paratropic and diatropic ring currents of the coronene moiety may, therefore, be modulated through the position of the heteroatom and, consequently, also the localized/delocalized behavior.

CCSD-CTOCD static dipole shielding polarizability for quantification of the chiral NMR effects in oxaziridine derivatives.

Chiral discrimination by nuclear magnetic resonance (NMR) spectroscopy might be achieved through the pseudo-scalar derived from the dipole shielding polarizability tensor. Coupled Cluster Singles and Doubles-Quadratic Response (CCSD-QR) calculations inside the continuous translation of the origin of the current density formalism have been carried out to determine the effects of basis set, electron correlation, and gauge translation on the determination of this magnitude in oxaziridine derivatives. Inclusion of electronic correlation is needed for adequately describing the pseudo-scalar for the heavier nuclei, making CCSD a rigorous and affordable method to compute these high order properties in medium-sized molecules. The observable magnitudes for chiral discrimination (produced RF voltage and required electric field) are calculated. Half of the considered molecules show values of the observable magnitudes near the lower limit for experimental detection. Nuclei (19)F, (31)P, and (79)Br produce the largest values of RF voltage (50-80 nV). Moreover, (31)P and (79)Br are the nuclei requiring smallest electric fields (3 MVm(-1)) to separate the NMR signals, being then suitable for both the techniques.

Delocalized currents without a ring of bonded atoms: strong delocalized electron currents induced by magnetic fields in noncyclic molecules.

Some noncyclic small molecules, electrically neutral or charged, sustain interatomic electronic currents in the presence of a stationary, spatially uniform magnetic field. The existence of fairly large delocalized electron flow is demonstrated in H2O, BH3, NH3, CH4, CH3-CH3, H3O⁺, CH3⁺, and NH4⁺, by plots of quantum mechanical current density. Convincing quantitative evidence is arrived at by current strengths, defined via a flux integral of the ab initio current density. Application of a simple ring current model shows that the delocalized current strengths account for the out-of-plane component of the magnetic shielding tensor along the symmetry axis. A definition of delocalized electron current as a current flowing along a closed loop containing three or more atoms is discussed.

On the determination of the diagonal components of the optical activity tensor in chiral molecules.

It is shown that the diagonal components of the mixed electric-magnetic dipole polarizability tensor, used to rationalize the optical rotatory power of chiral molecules, are origin independent, if they are referred to the coordinate system defined by the eigenvectors of the dynamic electric dipole polarizability, for a given value ω of the frequency of a monochromatic wave impinging on an ordered sample. Within this reference frame, the individual diagonal components of the mixed electric-magnetic dipole polarizability are separately measurable properties. The theoretical method is applied via a test calculation to the cyclic 1,2-M enantiomer of the dioxin molecule, using a large Gaussian basis set to estimate near Hartree-Fock values within a series of dipole length, velocity, and acceleration representations.

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.

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.

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) .

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).

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 ampère 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.

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.

Induced orbital paramagnetism and paratropism in closed-shell molecules.

Three-dimensional models of the quantum-mechanical current density induced by a uniform magnetic field in the electron cloud have been obtained for closed-shell systems BeH(-), BH, and CH(+), characterized by induced orbital paramagnetism, and in planar unsaturated hydrocarbons C(4)H(4) and clamped C(8)H(8), exhibiting pi paramagnetism. It is shown that, even for these paramagnetic systems, the paramagnetic contributions to magnetic susceptibilities and nuclear magnetic shielding, customarily taken into account in perturbation theory approaches, can formally be eliminated via the procedure of continuous transformation of the origin of the current density-paramagnetic zero. The definition of magnetic response properties can therefore be recast as a sum of two formally "diamagnetic" terms for any molecule, including systems showing strong induced orbital paramagnetism. It is shown that the paramagnetism in the compounds studied arises from the nodal topology of the electronic wave function. In particular, paratropic vortices circulate about stagnation lines at the intersection of nodal surfaces of the highest-occupied zero-order molecular orbital and corresponding first-order orbital.

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.

Relative Weights of σ and π Ring Currents in a Few Simple Monocycles.

By partitioning the bond current strength (current susceptibility) into plane symmetric and plane antisymmetric contributions, it is shown that 91% of the diatropic ring current of benzene is transported by the π electrons and the remaining non-negligible 9% is sustained by the σ electrons. In planar cyclooctatetraene 94% (6%) of the paratropic ring current is transported by the π (σ) electrons. In cyclopropane 95% (5%) of the diatropic ring current is transported by the σ (π-like) electrons. The 85% fraction of the diatropic ring current of Al4(2-) is transported by the σ valence electrons and 15% by the π valence electrons. In the nonaromatic borazine system the nitrogen-centered π electron circulations are surrounded by a weak diatropic "ring current" 6.5 times smaller than that of benzene.

Topology of magnetic-field induced electron current density in the cubane molecule.

A spatial model of the electronic current density induced in the cubane molecule by applying an external magnetic-field has been constructed employing quantum mechanical methods at the Hartree-Fock level of accuracy. The topological features of the current density vector field are described via a stagnation graph that shows the isolated points and the lines at which the current vanishes. Shielding density maps based on the differential Biot-Savart law, along with a collection of current density maps, explain magnetic shielding at hydrogen and carbon nuclei, and virtual shielding at ring and cage centers.

Relation between pi-electron localization/delocalization and H-bond strength in derivatives of omicron-hydroxy-schiff bases.

Detailed investigations of electronic effects taking place within the molecular system of o-hydroxy Schiff bases have been performed. The analysis of geometry, local and global aromaticity, selected AIM-based parameters, and finally, pi-electron currents induced in the systems under consideration have been performed on the basis of quantum chemical calculations at the B3LYP/6-311+G** level of theory. The relation between localization/delocalization of pi-electrons within the whole system has been described. It has been shown that the character of the bond which is common to the phenylic ring and the quasi-ring formed as a result of H-bond formation has a crucial impact on the strength of H-bonding. The strongest H-bonds can be observed for the systems in which the sequence of formally single and double bonds within the H-bridged quasi-ring enable a pi-electronic coupling. These observations indicate that pi-electron effects play a fundamental role in the stabilization of the hydrogen bridge within omicron-hydroxy Schiff bases. Analysis of pi-ring currents induced by a magnetic field perpendicular to the molecular plane of selected analyzed systems confirms these conclusions.

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.