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The closed-shell Hückel-London-Pople-McWeeny formalism for ring currents is extended to Aufbau configurations with open shells calculated as configurational averages. The method is applied to the non-Kekulean benzenoid known as the Clar goblet, recently synthesized on the Au(111) surface. Multiplicity of the ground state is a complication: for the Clar goblet, Hund's rule of maximum multiplicity implies a triplet whereas Ovchinnikov's rule implies a singlet. This disagreement has little effect on the predicted ring currents. Ring-current maps are calculated for the 36π dication, 40π dianion, and low-lying states of the 38π neutral, using Hückel-London and Hubbard-London models. All show twin diatropic perimeter currents on separate halves of the molecule. These are compared with ipsocentric pseudo-π and ab initio maps of induced π-current for closed-shell singlet configurations of dianion, dication, and neutral. Configurationally averaged Hückel-London calculations give a good account of the consistent diatropic ring currents in the Clar goblet for the three charge states.

Per iodoannulene molecules and ions C x I x q in planar geometry offer examples of systems with the potential for outer σ and inner π ring-current double aromaticity, given a sufficient overlap of tangential p σ -orbital manifolds on the large atoms of the outer cycle. Previous theoretical work indicated concentric diatropic currents in the dication C 6 I 6 2+ . Ab initio ipsocentric calculations support an account in terms of frontier-orbital selection rules for current contributions in C 6 I 6 2+ (and radical C 6 I 6 + , implicated in recent experimental work on the oxidation of per iodobenzene). A σ/π analogue of the annulene-within-an-annulene model is applied here to per iodo systems based on cyclooctatetraene. Model species C 8 I 8 q with charges q = 0, +1, +2, +4, -2 and structures constrained to a planar D 4 h symmetry exhibit maps with all combinations of σ/π con- and counter-rotation, comprising global σ ring currents on the iodine perimeter and central π ring currents on the carbocycle. All can be rationalized by the separate application of the tropicity selection rules to the two subsystems, whether in singlet or triplet states.

We model a molecular device as a molecule attached to a set of leads treated at the tight-binding level, with the central molecule described to any desired level of electronic structure theory. Within this model, in the absence of electron-phonon interactions, the Landauer-Büttiker part of the Meir-Wingreen formula is shown to be sufficient to describe the transmission factor of the correlated device. The key to this demonstration is to ensure that the correlation self-energy has the same functional form as the exact correlation self-energy. This form implies that nonsymmetric contributions to the Meir-Wingreen formula vanish, and hence, conservation of current is achieved without the need for Green's Function self-consistency. An extension of the Source-Sink-Potential (SSP) approach gives a computational route to the calculation and interpretation of electron transmission in correlated systems. In this picture, current passes through internal molecular channels via resonance states with complex-valued energies. Each resonance state arises from one of the states in the Lehmann expansion of the one-electron Green's function, hole conduction derived from ionized states, and particle conduction from attached states. In the correlated device, the dependence of transmission on electron energy is determined by four structural polynomials, as it was in the tight-binding (Hückel) version of the SSP method. Hence, there are active and inert conduction channels (in the correlated case, linked to Dyson orbitals) governed by a set of selection rules that map smoothly onto the simplest picture.

As a key diagnostic property of benzenoids and other polycyclic hydrocarbons, induced ring current has inspired diverse approaches for calculation, modeling, and interpretation. Grid-based methods include the ipsocentric ab initio calculation of current maps, and its surrogate, the pseudo-π model. Graph-based models include a family of conjugated-circuit (CC) models and the molecular-orbital Hückel-London (HL) model. To assess competing claims for physical relevance of derived current maps for benzenoids, a protocol for graph-reduction and comparison was devised. Graph reduction of pseudo-π grid maps highlights their overall similarity to HL maps, but also reveals systematic differences. These are ascribed to unavoidable pseudo-π proximity limitations for benzenoids with short nonbonded distances, and to poor continuity of pseudo-π current for classes of benzenoids with fixed bonds, where single-reference methods can be unreliable. Comparison between graph-based approaches shows that the published CC models all shadow HL maps reasonably well for most benzenoids (as judged by L 1 -, L 2 -, and L ∞ -error norms on scaled bond currents), though all exhibit physically implausible currents for systems with fixed bonds. These comparisons inspire a new combinatorial model (Model W) based on cycle decomposition of current, taking into account the two terms of lowest order that occur in the characteristic polynomial. This improves on all pure-CC models within their range of applicability, giving excellent adherence to HL maps for all Kekulean benzenoids, including those with fixed bonds (halving the rms discrepancy against scaled HL bond currents, from 11% in the best CC model, to 5% for the set of 18 360 Kekulean benzenoids on up to 10 hexagonal rings). Model W also has excellent performance for open-shell systems, where currents cannot be described at all by pure CC models (4% rms discrepancy against scaled HL bond currents for the 20112 non-Kekulean benzenoids on up to 10 hexagonal rings). Consideration of largest and next-to-largest matchings is a useful strategy for modeling and interpretation of currents in Kekulean and non-Kekulean benzenoids (nanographenes).

Derivative current-density maps are used to follow the changes in ring-current (and hence, on the magnetic criterion, the changes in aromaticity) with the Kekulé vibrations of the prototypical aromatic, antiaromatic, and nonaromatic systems of benzene, cyclooctatetraene (COT), and borazine. Maps are computed at the ipsocentric CHF/6-31G**//RHF/6-31G** level. The first-derivative map for benzene shows a growing-in of localized bond currents, and the second-derivative map shows a pure, paratropic "antiring-current", leading to the conclusion that vibrational motion along the Kekulé mode will reduce the net aromaticity of benzene, on average. For planar-constrained D(4h) COT, the Kekulé mode (positive for reduction of bond-length alternation) increases paratropicity at both first and second order, indicating an average increase in antiaromaticity with zero-point motion along this mode. On the ring-current criterion, breathing expansions of benzene and D(4h) COT reduce aromaticity and increase antiaromaticity, respectively.

Conjugated-circuit models for induced π ring currents differ in the types of circuit that they include and the weights attached to them. Choice of circuits for general π systems can be expressed compactly in terms of matchings of the circuit-deleted molecular graph. Variants of the conjugated-circuit model for induced π currents are shown to have simple closed-form solutions for linear polyacenes. Despite differing assumptions about the effect of cycle area, all the models predict the most intense perimeter current in the central rings, in general agreement with ab initio current-density maps. All tend to overestimate the rate of increase with N of the central ring current for the [N]polyacene, in comparison with molecular-orbital treatments using ipsocentric ab initio, pseudo-π, and Hückel-London approaches.

Aromatic benzene and nonaromatic borazine can be modeled as the end-points of a continuous process evolving through intermediate systems with fractional nuclear charges. Current-density maps show a smooth, linear progression in which the global diatropic π ring current weakens as localized diatropic lone-pair currents grow with the increase in charge difference. In contrast, the progression from antiaromatic (planarized) cyclooctatetraene to nonaromatic borazocine shows an initially persistent paratropic ring current with a sharper transition to the localized diatropic system. The different behaviors of aromatic and antiaromatic systems stem from the different orbital origins of diatropic and paratropic ring current, and both can be rationalized in terms of arguments based on π distortivity and electronegativity.

This paper presents a new procedure for identifying that part of the pi electronic energy of conjugated hydrocarbons which results from cyclic circulation of electrons around a ring. It first shows that one may calculate perturbatively the ground state energy of the Hückel Hamiltonian from a strongly localized Kekulé-type zero-order wave function. The contributions due to cyclic circulation of the electrons appear explicitly, in terms of the interatomic hopping integral t, at the second order in cyclobutadiene (where it is equal to -t (antiaromatic)) and at third order in benzene, where its value is 0.5t (aromatic). Conjugated isomers of benzene are also considered. The cyclic circulation contributions for an N-membered ring are shown to depend strongly on the molecular graph in which it is embedded. A general expression is found for the cyclic contribution to the pi energy of a ring, the Kekulé graph of which contains N double bonds alternating with N single bonds. It is the energy of the ring, plus the sum of the energies of the N subsystems that result from one double-bond removal, minus the sum of the energies of the N open systems that result from one single-bond cut. This new aromaticity index, ACE(MC), may be seen as the enthalpy of a hyperhomodesmotic chemical equation. In contrast to the index ACE(DC) previously defined from a double cut of the ring, the multiple-cut ACE(MC) exhibits the expected asymptotic disappearance of the cyclic energy as the ring size tends to infinity. In the multiple-cut approach, aromaticity persists in bond-alternating rings, but, in contrast to the total pi energy, the purely cyclic contribution tends to resist distortion. Extension of the approach to charged, branched and heterosubstituted rings are discussed, as well as its ab initio transcription.

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.

In an axial magnetic field, coronene, corannulene and kekulene support disjoint paratropic-hub and diatropic-rim ring currents. Bond-order and orbital arguments suggest [10,5]-coronene, C30H10, comprising 10 fused pentagons around a central decagon, as a system that should support an inverted diatropic-hub/paratropic-rim pattern of induced currents. The proposal is verified by ipsocentric ab initio mapping of currents, reproduced with the economical pseudo-pi method, thus validating a powerful toolkit for design of (induced) molecular magnets.