Standard Response Calculation of Electric Dipole Polarizability and Specific Optical Rotation Power Densities.

A time-dependent method  has been developed to solve the standard response equation for the calculation of dynamic  molecular property densities, endowed with the characteristic of being origin-invariant, entirely in the atomic orbital basis at both HF and DFT level of theory. The method has been tuned in particular for the calculation of origin-independent electric dipole polarizability density and specific rotation power density. Some demonstrations are given for the hexabenzocoronene molecule and the Tröger base.

Electronic Current Density Induced by Uniform Magnetic Fields in Clarenes.

Some planar and non-planar clarenes have been studied using maps of magnetically induced quantum-mechanical current density and tools from differential topology to assess their magnetic response in connection with recent results by Du and Wang. Bond current strengths have been computed to estimate quantitative measures. Isosurfaces of the divergence of induced Lorentz force density have been shown to provide useful additional criteria, especially in the case of non-planar clarenes.

Electronic current densities and origin-independent property densities induced by optical fields.

The interaction of a molecule with optical fields is customarily interpreted by means of induced time-dependent electric polarizabilities, magnetizabilities and mixed electric-magnetic polarizabilities. In general, these properties can be rationalized by integrals of density functions formulated in terms of induced charge and current densities. In this perspective, we focus on what has been done so far at the theoretical level, and on what can be expected to be unveiled from the topological study of suitable density functions, endowed with the fundamental requirement of origin invariance. Densities characterized by such a property can be integrated all over the configuration space to obtain electric dipole polarizability and optical rotatory power. Corresponding maps visualize domains mainly involved in the molecular response. The diagonal components of origin-independent density tensor functions that, on integration, yield corresponding electric dipole polarizability tensor of benzene, naphthalene, phenanthrene and ovalene, have been computed, confirming the ubiquitous presence of counter-polarization regions in the proximity of the atomic nuclei. They are associated with toroidal electron currents, induced by time derivative of the electric field of impinging radiation. Electron (de)localization in these systems is readily observed and estimated. The optical rotation density of the carbonyl chromophore is studied in detail. Its essential feature is the separation in quadrants of alternating sign of density about the CO bond. The presence of an extrachromophoric perturbation determines asymmetry in the extension of the quadrant distribution, thus causing optical rotation.

Methylene-Bridged [6]-, [8]-, and [10]Cycloparaphenylenes: Size-Dependent Properties and Paratropic Belt Currents.

Cycloparaphenylenes (CPPs) and carbon nanobelts (CNBs) represent some of the most iconic cyclic molecular nanocarbons in recent chemistry owing to their unique properties derived from rigid, strained, and cyclic π-conjugated systems. In the last decade, the synthesis of various sizes of CPPs and CNBs has been achieved that allowed not only for investigating their size-dependent properties and strategically using such properties in various applications but also understanding the fundamental features of cyclic π-conjugated systems and molecular nanocarbons in general. Herein, we report on the synthesis, size-dependent properties, and paratropic belt currents of methylene-bridged [n]cycloparaphenylenes ([n]MCPP, n = 6, 8, 10). [8]MCPP and [10]MCPP were synthesized by the same strategy we developed for [6]MCPP synthesis. With readily available ethoxy-substituted pillar[8]arene and pillar[10]arene as precursors, [8]MCPP and [10]MCPP were successfully synthesized in three steps consisting of de-ethylation, triflation, and nickel-mediated aryl-aryl coupling. The structural and electronic properties of MCPPs were investigated by nuclear magnetic resonance analyses, absorption/fluorescence measurements, X-ray crystallographic analyses, and computational studies, revealing their interesting size-dependent properties. The differences in the size dependency between MCPPs and CPPs reflect the belt-form features of MCPPs, namely, methylene-bridging effects on MCPPs. Moreover, an interesting paratropic belt current along the MCPP backbone has been uncovered both experimentally and theoretically. The 1H NMR chemical shifts of MCPPs confirmed the presence of a paratropic belt current, whose strength rapidly decreases with increasing nanobelt size.

On the computation of frequency-dependent molecular magnetizabilities via dynamical charge and current electron densities.

In the computation of molecular dynamic magnetizabilities and magnetic dipole moments, three different reference points are required: (i) origin of coordinate system, (ii) origin of vector potential A , and (iii) origin of multipole expansion. This study shows that methods relying on continuous translation of origin of the current density I B r ω t induced by optical magnetic fields provide an effective solution to the problem of choices (i) and (ii), in that they yield origin independent I B within the algebraic approximation, for any basis set. Frequency-dependent magnetizabilities are also invariant with respect to (iii), as a consequence of symmetry, for a number of molecular point groups. In molecules of lower symmetries, computed magnetizabilities depend on origin of the multipole expansion. Large basis set computations carried out for water, ammonia, methane, ethane, ethylene, boranylborane, and hydroxilamine, at the DFT level, have been reported to document these statements. A comparison is made for results obtained within the conventional common origin approach for static magnetic field. Sum rules for invariance of computed properties are discussed. Representations of streamlines and stagnation graphs of dynamical current density vector field induced in the water molecule by monochromatic waves of four frequencies are displayed.

Size-Reduced Basis Set Calculation of Accurate Isotropic Nuclear Magnetic Shieldings Using CTOCD-GRRO and GPRO Methods in Amino Acids and Oligopeptides.

The origin-independent magnetically induced CTOCD-GRRO and -GPRO (after continuous transformation of the origin of the current density-gradient of ρ and gradient of a power of ρ) current densities are shown to vary linearly with respect to their own defining α and ß parameters. The same is reflected in the connected magnetic properties, in particular the magnetic shielding. This is exploited to find values for α and ß that, using small basis sets, provide isotropic nuclear magnetic shieldings matching an accurate prediction, chosen as the complete basis set limit. An application to the 20 naturally occurring amino acids shows that different nuclei require different values of the parameters, which have been determined at the BHandHLYP/6-31+G(d,p) level with or without consideration of diversified chemical environments. Using CTOCD-GRRO and -GPRO methods, equipped with the optimized parameters at this low-cost level of calculation, 1H, 13C, 15N, and 17O magnetic shielding constants in glutathione, ophthalmic acid, and thyrotropin-releasing hormone are predicted with nearly the same accuracy as that of much more expensive calculations.

Synthesis and Stereochemical Characterization of a Novel Chiral α-Tetrazole Binaphthylazepine Organocatalyst.

A novel α-tetrazole-substituted 1,1'-binaphthylazepine chiral catalyst has been synthesized and its absolute configuration has been determined by DFT computational analysis of the vibrational circular dichroism (VCD) spectrum of its precursor. The VCD analysis, carried out through the model averaging method, allowed to assign the absolute configuration of a benzylic stereocenter in the presence of a chiral binaphthyl moiety. The 1,1'-binaphthylazepine tetrazole and the nitrile its immediate synthetic precursor, have been preliminarily tested as chiral organocatalysts in the asymmetric intramolecular oxa-Michael cyclization of 2-hydroxy chalcones for the synthesis of chiral flavanones obtaining low enantioselectivity.

On the JAP Method for the Indirect Determination of Delocalized Currents from Experimental Chemical Shifts.

The JAP model (after Jirásek, Anderson, and Peeks) to retrieve global current strengths from experimental 1 H chemical shifts has been tested with DFT computations. Both global and local tropicities are correctly predicted in most cases and the quantitative agreement is overall fair. An extension of the model is found to give improvement in an exemplary critical case, where the global delocalized current is negligible and the current density map is dominated by local currents.

Magnetic Characterization of the Infinitene Molecule.

The origin-independent current density induced by a perpendicular magnetic field in the infinitene molecule has been calculated, confirming the recently presented result by Orozco-Ic et al. ( Phys. Chem. Chem. Phys. 2022, 24, 6404-6409) of two disjointed global current pathways along the edges formed by 24 carbon atoms having the form of the infinity symbol. The current strength has been assessed along the C-C bonds forming the two separate circuits, whose particular shape provides a diamagnetic exaltation which is only 73% of the expected value for this aromatic molecule. Through space currents have been found along the bond paths determined by the electron density gradient, whose strength is 10% that of the aromatic benzene ring current. It is shown that the pair of high-field 1H NMR experimental signals carry the signature of the two global currents, which are counterrotating inside the fjord regions with respect to the rim of the coronene subunits.

Origin independent current density vector fields induced by time-dependent magnetic field. I. The LiH molecule.

The electronic current density, induced in a molecule by the optical magnetic field associated with a frequency-dependent monochromatic plane wave, assumed to be spatially uniform within the electric quadrupole approximation, has been studied by using a theoretical method based on a continuous translation of its origin. The induced electronic current density vector field designated by this procedure, invariant of the origin for any point of the molecular domain, is obtained via a computational scheme, formally annihilating the diamagnetic contribution of the conventional common-origin approach based on perturbation theory. In a preliminary application of the theoretical methods outlined in the present work, the simple molecule of lithium hydride has been investigated. Particular attention has been paid to the structure of induced electronic current density for several values of the magnetic field frequency by investigating equilibrium points of four different types, organized in stagnation lines, which constitute its stagnation graph, i.e., a topological fingerprint of the vector field conveying complete information in the condensed form, to verify the fulfillment of fundamental requirements, e.g., the continuity equation and the Poincaré-Hopf theorem on spherical and toroidal surfaces.

Dynamic toroidizability as ubiquitous property of atoms and molecules in optical electric fields.

The continuous search for metamaterials with special properties, suitable for new technological applications, is presently being driven by a preceding theoretical development, which took place after the introduction of new physical entities, anapole and a family of toroidal multipoles, having a border in common with those considered in the more familiar electric and magnetic multipole expansions. The related concept of toroidization, i.e., toroidal moment per unit volume, has been advocated in analogy to electric polarization and magnetization operated by electromagnetic fields and should be considered on the same footing regarding its relevance and practicality for understanding certain properties, e.g., ferrotoroidicity in condensed matter physics, and for rationalizing the behavior of charge-current distributions that neither radiate nor interact with external fields in classical and quantum electrodynamics. Toroidizability, i.e., the ability of sustaining toroidal moments, can also be defined by an analogy with electric polarizability and magnetizability. The present study shows that such a property is general and characterizes atoms and molecules and that the optical electric field of a light beam induces an oscillating anapole moment, i.e., the superposition of toroidal moment with an electric dipole moment. However, values of anapole polarizabilities induced by monochromatic light, estimated by time-dependent perturbation theory for rare gas atoms and a few molecules, are quite small and possibly hard to detect experimentally.

Origin-Independent Densities of Static and Dynamic Molecular Polarizabilities.

The notion of the electric dipole polarizability density function of atoms and molecules has been considered. The current density induced by the time derivative of the electric field of monochromatic light allows for a new definition of the electric dipole polarizability density, which is translationally invariant. This translational invariance provides the physical meaning that was lacking in previous defined polarizability densities. The new polarizability density has been implemented at the TD-DFT level of theory. The origin independence has been proven in silico to hold regardless of the basis set size. Two emblematic molecules, i.e., CO and N2, which in many respects display similar electric response, have been studied in detail. The substantial differences, which have been highlighted in the topology of the parallel and perpendicular polarizability density tensor components of CO and N2, are grossly hidden by compensation, when integration is carried out to get the molecular properties.

Assessment of the performance of DFT functionals in the fulfillment of off-diagonal hypervirial relationships.

Off-diagonal hypervirial relationships, combined with quantum mechanical sum rules of charge-current conservation, offer a way to test electronic excited-state transition energies and moments, which does not need any external reference. A number of fundamental relationships were recast into absolute deviations from zero, which have been used to assess the performance of some popular DFT functionals. Extended TD-DFT calculations have been carried out for a pool of molecules chosen for this purpose, adopting a large basis set to ensure high quality results. A partial agreement with previous benchmarks is observed.

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.

Program Package for the Calculation of Origin-Independent Electron Current Density and Derived Magnetic Properties in Molecular Systems.

We present SYSMOIC, a program package for the calculation of the origin-independent current density induced at first order by an external magnetic field in planar and nonplanar molecular systems. Origin independence is obtained adopting the continuous transformation of the origin of the current density method, implemented at both density functional theory (DFT) and Hartree-Fock (HF) levels. Expansion coefficients for perturbed and unperturbed molecular orbitals, over basis sets containing up to m-type Gaussian functions, can be calculated by the package itself or obtained from a Gaussian calculation. A number of different functionalities presented so far in the literature that are connected to the induced current, such as current density maps for any orientation of the inducing magnetic field, net bond current strengths, stagnation graphs, magnetic shielding densities, vorticities, and anisotropies, are now made available all together in a single multiplatform package installation.

The molecular electronic structure revealed by the magnetically induced Lorentz force density.

The energy change occurring by perturbing a closed-shell molecule, either fixed in space or randomly tumbling, by a uniform external magnetic field has been expressed in terms of the magnetically induced Lorentz force density, which, at difference with previously introduced energy-based quantities, is independent of the point-of-view. The divergence of the isotropically averaged magnetically induced Lorentz force density allows for a local definition of diamagnetic and paramagnetic responses and enables us to distinguish aromatic, antiaromatic, and non-aromatic molecules, as shown by calculations on benzene, cyclooctatetraene, and borazine. The isotropically averaged magnetically induced Lorentz force density turns out very similar to the gradient of the electron density. This similarity is justified in terms of the current knowledge on the current density. A way is opened for an insightful and simplified topological characterization of molecular electronic structure via the magnetic response.

Disentangling the Contributions to the Proton Magnetic Shielding in Carbon Nanohoops and Nanobelts: Evidence for a Paratropic Belt-Current.

The proton NMR magnetic shieldings of the recently synthesized D3d isomers of methylene-bridged [6]cycloparaphenylene (MB[6]CPP) and [12]cyclophenacene hide in themselves the effect of a global paratropic current around the nanobelts, which is induced by a magnetic field parallel to the main symmetry axis of the molecules. The effect is particularly pronounced for the methylene protons of MB[6]CPP, especially for those facing inside the nanobelt. The small experimental chemical shift difference of only 0.2 ppm is incompatible with the separation of the signals caused by the belt curvature, which, by itself, is calculated to be larger than 1 ppm, with both signals shifted upfield with respect to the position detected for the nanobelt. A careful dissection of the proton magnetic shielding in terms of molecular orbital contributions, has permitted a quantitative assessment of the genuine effect on each different proton caused by a substantial paratropic belt-current, which brings all the signals in nice agreement with the experimental spectra.

Reversal of Clar's Aromatic-Sextet Rule in Ultrashort Single-End-Capped [5,5] Carbon Nanotubes.

The three-fold HOMO-LUMO gap oscillation, typical of finite length armchair carbon nanotubes (CNT), has a major effect on the magnetic response of ultrashort, single-end-capped [5,5] carbon nanotubes to a perturbing magnetic field parallel to the main symmetry axis. For the CNT's containing 40, 70, and 100 carbon atoms, for which 100 % of the C=C double bonds can be grouped into aromatic-sextets, i. e., fully or complete Clar networks, large paratropic (antiaromatic) global circulations around the cylindrical axis are predicted at the DFT level of calculation. Local and semi-global diatropic (aromatic) currents of strengths not larger than that of the benzene molecule are determined for a perpendicular perturbing magnetic field. CNTs of intermediate lengths do not display this enhanced antiaromatic response. The paratropic current flow clearly shows that these complete Clar networks can be viewed as stacked cycloparaphenylene belts, each providing a double 4n annulene circuit as a consequence of the quinoidal resonance structure that results from their closure. Paradoxically, the fully aromatic Clar structure itself is responsible for the enhanced global antiaromaticity.

Design of annulene-within-an-annulene systems by the altanisation approach. A study of altan-[n]annulenes.

The altanisation strategy, devised to design molecules with large and paratropic perimeter circulations, is applied to the family of [n]annulenes to give, altan-[n]annulenes, i.e. [n,5]coronenes. Analytical expressions are obtained for the eigenvalues of the Hückel Hamiltonian for altan-[n]annulenes, and used in conjunction with selection rules derived from the ipsocentric approach to predict patterns of global ring current in these systems. Density-functional calculations performed on seven altan-[n]annulenes, three neutral and four charged, give current-density maps in essential agreement with the predictions obtained at the unperturbed Hückel level. All but one of the systems show patterns with the tropicities expected for isolated annulenes, in line with the altanisation concept. The apparent exception is altan-[11]annulene-, the only singlet system with a well defined open-shell character in the studied set. The key role of open-shell character can be accommodated by appropriate choice of the occupation numbers of the initial Hückel molecular orbitals, where the anion altan-[11]annulene- is considered as an [11]annulene inside the [22]annulene anion.

NICS-XY-Scan Predictions of Local, Semi-Global, and Global Ring Currents in Annulated Pentalene and s-Indacene Cores Compared to First-Principles Current Density Maps.

The magnetic response of a set of 20 molecules, their dications and their dianions has been studied by the NICS-XY method, and the results have been compared with first principles current density maps. The molecules have been built from pentalene and s-indacene by single and double annulation of cyclobutadiene, benzene and benzocyclobutadiene in an alternate fashion on both sides of the molecules. The prediction of tropicities obtained by the NICS-XY method are overall consistent with the current density maps. A literal code, developed to give a compact description of the tropicities of currents flowing around rings and bonds cut by the scan trajectory shows that, in most cases, the NICS-XY method leads to an exact match with the current density analyses. Mismatches are generally due to small circulations out of the scan trajectory, and they do not correspond to misinterpretations of the overall tropicities. The dataset provides several cases where the prediction of the overall antiaromatic/aromatic response by the 4n/4n+2 count of π electrons fails.