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.     

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.     

Calculation of current densities using gauge-including atomic orbitals

A method for calculating the various components of the magnetically induced current-density tensor using gauge-including atomic orbitals is described. The method is formulated in the framework of analytical derivative theory, thus enabling implementation at the Hartree-Fock self-consistent-field (HF-SCF) as well as at electron-correlated levels. First-order induced current densities have been computed up to the coupled-cluster singles and doubles level (CCSD) augmented by a perturbative treatment of triple excitations [CCSD(T)] for carbon dioxide and benzene and up to the full coupled-cluster singles, doubles, and triples (CCSDT) level in the case of ozone. The applicability of the gauge including magnetically induced current method to larger molecules is demonstrated by computing first-order current densities for porphin and hexabenzocoronene at the HF-SCF and density-functional theory level. Furthermore, a scheme for obtaining quantitative values for the induced currents in a molecule via numerical integration over the current flow is presented. For benzene, a perpendicular magnetic field induces a (field dependent) ring current of 12.8 nA T(-1) at the HF-SCF level using a triple-zeta basis set augmented with polarization functions (TZP). At the CCSD(T)/TZP level the induced current was found to be 11.4 nA T(-1). Gauge invariance and its relation to charge-current conservation is discussed.     

Induced currents and electron counting in aromatic boron wheels: B8(2-) and B9-)

The newly discovered atom-centered polygonal wheels B8(2-) and B9- are predicted to show ring currents characteristic of aromatic systems. Ipsocentric mapping of induced current density for both molecules attributes a pi diatropic current to the four electrons of the doubly degenerate pi HOMO and a sigma diatropic current to the four electrons of the doubly degenerate sigma HOMO, each orbital pair having an available transition to corresponding LUMO orbitals in which the angular node count increases by one. Thus, on the magnetic criterion, B8(2-) and B9- are each both pi- and sigma-aromatic as a consequence of the nodal properties of the frontier orbitals of the pi- and sigma-stacks.     

Aromatic pathways of porphins, chlorins, and bacteriochlorins

Magnetically induced current densities have been calculated for free-base porphynoids using the gauge including magnetically induce current (GIMIC) method. Numerical integration of the current density passing selected chemical bonds yields current pathways and the degree of aromaticity according to the magnetic criterion. The ring-current strengths of the porphins, chlorins, and bacteriochlorins are 1.5-2.5 times stronger than for benzene. The calculations show that the 18π [16]annulene inner cross is not the correct picture of the aromatic pathway for porphyrins. All conjugated chemical bonds participate in the current transport independently of the formal number of π electrons. The ring current branches at the pyrrolic rings taking both the outer and the inner route. The NH unit of the pyrrolic rings has a larger resistance and a weaker current strength than the pyrroles without inner hydrogens. The traditional 18π [18]annulene with inactive NH bridges is not how the ring-current flows around the macroring. The porphins have the strongest ring current of ca. 27 nA/T among the investigated porphynoids. The current strengths of the chlorins and bacteriochlorins are 19-24 nA/T depending on whether the ring current is forced to pass an NH unit or not. The current strengths of the 3-fold and 4-fold β-saturated porphynoids are 13-17 nA/T, showing that the inner-cross 18π [16]annulene pathway is not a preferred current route.     

Probing the aromaticity of the [(H(t)Ac)3(μ2-H)6], [(H(t)Th)3(μ2-H)6],(+), and [(H(t)Pa)3(μ2-H)6] clusters

In this study we report about the aromaticity of the prototypical [(H(t)Ac)(3)(μ(2)-H)(6)], [(H(t)Th)(3)(μ(2)-H)(6)](+), and [(H(t)Pa)(3)(μ(2)-H)(6)] clusters via two magnetic criteria: nucleus-independent chemical shifts (NICS) and the magnetically induced current density. All-electron density functional theory calculations were carried out using the two-component zeroth-order regular approach and the four-component Dirac-Coulomb Hamiltonian, including scalar and spin-orbit relativistic effects. Four-component current density maps and the integration of induced ring-current susceptibilities clearly show that the clusters [(H(t)Ac)(3)(μ(2)-H)(6)] and [(H(t)Th)(3)(μ(2)-H)(6)](+) are non-aromatic whereas [(H(t)Pa)(3)(μ(2)-H)(6)] is anti-aromatic. However, for the thorium cluster we find a discrepancy between the current density plots and the classification through the NICS index. Our results also demonstrate the increasing influence of f orbitals, on bonding and magnetic properties, with increasing atomic number in these clusters. We think that the enhanced electron mobility in [(H(t)Pa)(3)(μ(2)-H)(6)] is due the significant 5f character of its valence shell. Also the participation of f orbitals in bonding is the reason why the protactinium cluster has the shortest bond lengths of the three clusters. This study provides another example showing that the magnetically induced current density approach can give more reliable results than the NICS index.     

A dissected ring current model for assessing magnetic aromaticity: a general approach for both organic and inorganic rings

A model based on classical electrodynamics is used to measure the strength of ring currents of different molecular orbitals, i.e., σ- and π-orbitals, and characteristics of ring current loops, i.e., ring current radii and height of current loops above/below the ring planes, among a number of organic as well as inorganic molecules. For the π-current, the present model represents an improvement of previous approaches to determine ring current intensity. It is proven that the present model is more precise than previous models as they could not explain presence of the minimum in the plot of NICS(πzz) versus distance close to the ring plane. Variations in the charge of molecules and the types of constituent atoms of each species affect the ring current radii of both σ- and π-current loops as well as the height of π-current loops above/below the ring plane. It is suggested that variation in the distribution of the one-electron density in different systems is the main source of differences of the ring current characteristics.     

The gauge including magnetically induced current method

An overview of applications of the recently developed gauge including magnetically induced current method (GIMIC) is presented. The GIMIC method is used to obtain magnetically induced current densities in molecules. It provides detailed information about electron delocalization, aromatic character, and current pathways in molecules. The method has been employed in aromaticity studies on hydrocarbons, complex multi-ring organic nanorings, M?bius twisted molecules, inorganic and all-metal molecular rings and open-shell species. Recent studies on hydrogen-bonded molecules indicate that GIMIC can also be used to estimate hydrogen-bond strengths without fragmentation of the system. Preliminary results are presented on the applicability of GIMIC for investigating current transport in molecules attached to clusters simulating molecular conductivity measurements. Advantages and limitations of the GIMIC method are reviewed and discussed.     

Why the hydration energy of Au+ is larger for the second water molecule than the first one: skewed orbitals overlap

Using molecular-orbital analysis, we have elucidated the quantum-chemical origin of the intriguing phenomena in sequential hydration energies of the gold cation, which is known to be the most conspicuous among all transition metals. The hydration energy of Au+ with the second water molecule is found to be much larger than that with the first water molecule. Owing to the large relativistic effect of gold (i.e., significant lowering of the 6s orbital energy and significant raising of the 5d orbital energy), the highest occupied molecular orbital of the hydrated gold cation has a large portion of the 6s orbital. As the electron density of the 6s orbital populates in a large outer spherical shell far off the gold nucleus, the p orbitals (or sp hybridized lone-pair orbitals) of the water molecules are able to overlap with the outer part of the 6s orbital in the dihydrated gold cation, resulting in the unusual skewed overlap of p-6s-p orbitals (not the atom-to-atom bond overlap). No previous molecular-orbital analysis has reported this peculiar skewed orbitals overlap. Since this skewed orbitals overlap is saturated with two water molecules, this property is responsible for the low coordination number of the gold ion.     

Tuning the Structure and Properties of N-doped Positively Charged Polycyclic Aromatic Hydrocarbons

A detailed study of the geometry, aromatic character, electronic and magnetic properties for a series of positively charged N-doped polycyclic aromatic hydrocarbons (PAHs) was performed. Magnetic properties of the examined molecules were analyzed by means of the magnetically induced current density calculated using the diamagnetic-zero version of the continuous transformation of origin of current density (CTOCD-DZ) method. The comparative study of the local aromaticity of the studied molecules was performed using several different indices: energy effect (ef), harmonic oscillator model of aromaticity (HOMA) index, six centre delocalization index (SCI) and nucleus independent chemical shifts (NICS). The presence of N-atoms in the inner rings was found to cause a planarity distortion in the studied N-doped systems. The geometric changes and charged nature of the studied N-doped systems do not significantly influence the current density and the local aromaticity distribution in comparison with the corresponding parent benzenoid hydrocarbons. The present study demonstrates how quantum chemical calculations can be used for rational design of novel PAHs and for fine tuning of their properties.     

Ring currents in boron and carbon buckyballs, B80 and C60

The ipsocentric method at the coupled Hartree-Fock level is used for the calculation of magnetically induced ring currents in the boron buckyball B(80), for both I(h) and distorted T(h) geometries. A close similarity between the current patterns in boron and carbon buckyballs is noted, but with a higher current density in B(80). Paratropic currents on the pentagons are predominant in the boron buckyball, and the central NICS value is positive. These observations support the conclusion that B(80) should be considered (weakly) anti-aromatic. The largest orbital contributions to the ring currents in both molecules are identified and related to specific excitations in the frontier orbital region.     

Interplay of Hückel and Möbius Aromaticity in Metal-Metal Quintuple Bonded Complexes of Cr,Mo, and W with Amidinate Ligand: Ab initio DFT and Multireference Analysis**

The aromaticity of metal-metal quintuple bonded complexes of the type M₂L₂ (M=Cr, Mo, and W; L=amidinate) are studied employing gauge including magnetically induced ring current (GIMIC) analysis and electron density of delocalized bonds (EDDB). It is found that the complexes possess two types of aromaticity: i) Hückel aromaticity through delocalization of ligand π electrons with metal-metal δ-bond-forming 6 conjugated electrons (4π and 2δ) ring; ii) Craig-Möbius aromaticity through delocalization of π electrons of both the ligands with metal d-orbitals in Craig type orientation forming 10π electrons ring with a double twist. Extended transition state natural orbital chemical valence (ETS-NOCV) and canonical molecular orbital natural chemical shielding (CMO-NCS) analysis confirm the Craig-Möbius type arrangement of the orbitals. Furthermore, the unprecedented Hückel and Möbius type aromaticity is confirmed from the plot of the current pathways using 3D line integral convolution (3D-LIC) plots. The metal-metal bond order also increases down the group as justified from the complete active space self-consistent field (CASSCF) analysis. Due to an increase in the π and δ electron conjugation, both the Hückel and Möbius aromaticity increase down the group.     

Full spectral decomposition of ring currents

Within the ipsocentric method for calculation of molecular magnetic response, projection of perturbed orbitals onto the virtual orbital space allows partition of induced current density into contributions from individual virtual excitations between occupied and unoccupied orbitals, enabling detailed assignment of the origin of currents in, e.g., benzene, cyclooctatetraene, borazine, coronene, and corannulene. Whereas delocalized currents in benzene and planar cyclooctatetraene are described by transitions within the valence space, localized currents in the borazine pi system involve excitations outside the valence space.     

Magnetizability tensors from auxiliary density functional theory

The working equations for the calculation of the magnetizability tensor in the framework of auxiliary density functional theory with gauge including atomic orbitals (ADFT-GIAO) are derived. Unlike in the corresponding conventional density functional theory implementations the numerical integration of the GIAOs is avoided in ADFT-GIAO. Our validation shows that this simplification has no effect on the accuracy of the methodology. As a result, a reliable and efficient implementation for the calculation of magnetizabilities of systems with more than 1000 atoms and 14 000 basis functions is presented.     

Calculation of Verdet constants with time-dependent density functional theory: implementation and results for small molecules

We report the implementation of a method by which to calculate Verdet constants for molecules. The method is based on gauge-including atomic orbitals (GIAOs) and density functional theory. Calculations based on this method afford magneto-optical rotations of the right magnitude for the molecules H2, N2, CO, HF, CH4, C2H2, H2O, and CS2. The results are in satisfactory agreement with experiment. We investigate the dependency of the results on the gauge origin if GIAOs are not chosen, the convergence of the results with the size of the basis set for AOs and GIAOs, and for H2O and CS2 a comparison of gas-phase and liquid phase values. For the small molecules studied here, large polarized basis sets with diffuse functions are required to obtain well converged results. The use of an asymptotically correct Kohn-Sham potential is advantageous.     

Gauge invariant calculations of nuclear magnetic shielding constants using the continuous transformation of the origin of the current density approach. II. Density functional and coupled cluster theory

The quantum mechanical current density induced in a molecule by an external magnetic field is invariant to translations of the coordinate system. This fundamental symmetry is exploited to formally annihilate the diamagnetic contribution to the current density via the approach of "continuous transformation of the origin of the current density-diamagnetic zero" (CTOCD-DZ). The relationships obtained by this method for the magnetic shielding at the nuclei are intrinsically independent of the origin of the coordinate system for any approximate computational scheme relying on the algebraic approximation. The authors report for the first time an extended series of origin-independent estimates of nuclear magnetic shielding constants using the CTOCD-DZ approach at the level of density functional theory (DFT) with four different types of functionals and unrelaxed coupled cluster singles and doubles linear response (CCSD-LR) theory. The results obtained indicate that in the case of DFT the procedure employed is competitive with currently adopted computational methods allowing for basis sets of gauge-including atomic orbitals, whereas larger differences between CTOCD-DZ and common origin CCSD-LR results are observed due to the incomplete fulfillment of hypervirial relations in standard CCSD-LR theory. It was found furthermore that the unrelaxed CCSD-LR calculations predict larger correlation corrections for the shielding constants of almost all nonhydrogen atoms in their set of molecules than the usual relaxed energy derivative CCSD calculations. Finally the results confirm the excellent performance of Keal and Tozer's third functional, in particular, for the multiply bonded systems with a lot of electron correlation, but find also that the simple local density functional gives even better results for the few singly bonded molecules in their study where correlation effects are small.     

Synthesis,Aromaticity and Photophysical Behaviour of Ferrocene‐ and Ruthenocene‐Appended Semisynthetic Chlorin Derivatives

Two novel synthetic strategies to covalently link a metallocene electron‐donor unit to a chlorin ring are presented. In one approach, pyropheophorbide a is readily converted into its 13¹‐ferrocenyl dehydro derivative by nucleophilic addition of the ferrocenyl anion to the 13¹‐carbonyl group. In another approach, the corresponding 13¹‐pentamethylruthenocenyl derivative is synthesised from 13¹‐fulvenylchlorin by a facile ligand exchange/deprotonation reaction with the [RuCp*(cod)Cl] (Cp*=pentamethylcyclopentadienyl; cod=1,5‐cyclooctadiene) complex. The resulting metallocene–chlorins exhibit reduced aromaticity, which was unequivocally supported by ring‐current calculations based on the gauge‐including magnetically induced current (GIMIC) method and by calculated nucleus‐independent chemical shift (NICS) values. The negative ring current in the isocyclic E ring suggests the antiaromatic character of this moiety and also clarifies the spontaneous reactivity of the complexes with oxygen. The oxidation products were isolated and their electrochemical and photophysical properties were studied. The ruthenocene derivatives turned out to be stable under light irradiation and showed photoinduced charge transfer with charge‐separation lifetimes of 152–1029 ps.