Conjugated Circuits Model Research Articles

Partitioning Hückel–London Currents into Cycle Contributions

Ring-current maps give a direct pictorial representation of molecular aromaticity. They can be computed at levels ranging from empirical to full ab initio and DFT. For benzenoid hydrocarbons, Hückel–London (HL) theory gives a remarkably good qualitative picture of overall current patterns, and a useful basis for their interpretation. This paper describes an implemention of Aihara’s algorithm for computing HL currents for a benzenoid (for example) by partitioning total current into its constituent cycle currents. The Aihara approach can be used as an alternative way of calculating Hückel–London current maps, but more significantly as a tool for analysing other empirical models of induced current based on conjugated circuits. We outline an application where examination of cycle contributions to HL total current led to a simple graph-theoretical approach for cycle currents, which gives a better approximation to the HL currents for Kekulean benzenoids than any of the existing conjugated-circuit models, and unlike these models it also gives predictions of the HL currents in non-Kekulean benzenoids that are of similar quality.

Ring-Current Maps for Benzenoids: Comparisons, Contradictions, and a Versatile Combinatorial Model.

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 L1-, L2-, 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).

The determination of resonance energy in conjugated benzenoids: A combined experimental and theoretical approach

The conjugated circuits model of aromaticity in benzenoids is reconsidered. Values for the parameters R1, R2, and R3 are rederived from experimental enthalpies of formation. It is shown that the value of R3 depends on the shape as well as the size of a 14-cycle. R3 values are found to obey the relation R3pyrene>R3anthracene>R3phenanthrene.

Perimeter ring currents in benzenoids from Pauling bond orders.

It is shown that the ring currents in perimeter hexagonal rings of Kekulean benzenoids, as estimated within the Randić conjugated-circuit model, can be calculated directly without tedious pairwise comparison of Kekulé structures or Kekulé counting for cycle-deleted subgraphs. Required are only the Pauling bond orders of perimeter bonds and the number of Kekulé structures of the benzenoid, both readily available from the adjacency matrix of the carbon skeleton. This approach provides easy calculation of complete current maps for benzenoids in which every face has at least one bond on the perimeter (as in the example of cata-condensed benzenoids), and allows qualitative evaluation of the main ring-current contributions to (1)H chemical shifts in general benzenoids. A combined Randić-Pauling model for correlation of ring current and bond length through bond order is derived and shown to be consistent with resilience of current under bond alternation.

Equiaromatic benzenoids: Arbitrarily large sets of isomers with equal ring currents

A notion of equiaromaticity is defined: two polycyclic systems are equiaromatic if their rings can be put into one-to-one correspondence such that corresponding π ring currents are equal. In conjugated-circuit models, amongst other families, all isomeric unbranched catafused benzenoids sharing the same sequence of straight-chain lengths are predicted to be equiaromatic. In particular, all fibonacenes with equal numbers of hexagonal rings, from zig-zag to helicenoid isomers, are equiaromatic according to all conjugated-circuit models. Test calculations with small fibonacenes indicate that equiaromaticity is retained as a qualitative trend in the simplest molecular-orbital treatment of induced current.

Capturing the elusive aromaticity of bicalicene

The ring-current aromaticity of the bicalicene molecule arises, in spite of the 16 π carbon perimeter, from strong local diatropic circulations on the two pentagonal rings, as shown by current-density maps computed at the ipsocentric RHF/6-311G** and DFT/6-311G** levels of theory. Conjugated-circuit models cannot capture this pattern of circulation as it arises from 'ionic' contributions in a valence-bond picture. Canonical molecular-orbital analysis reveals a cancellation of paratropic and diatropic frontier-orbital contributions, which explains the difficulties that Hückel-based models have in producing qualitatively correct current-density maps for this molecule. Other measures of aromaticity reflect, to different extents, the dominance of the 'tetraionic' contribution to the aromaticity of this species.

Three Contra-Rotating Currents from a Rational Design of Polycyclic Aromatic Hydrocarbons: altan-Corannulene and altan-Coronene

Both the ab initio expression of the current density within the ipsocentric approach and conjugated circuit models indicate that placing an unsaturated hydrocarbon inside a [4n]annulene, in such a way that outgoing C-H bonds are substituted by C-C bonds to alternating carbon atoms of the annulene, leads to an homologue altan-molecule whose perimeter is expected to preserve the paratropic circulation of the parent annulene. Computations of current on the novel altan-corannulene and altan-coronene revealed unprecedented patterns of three contra-rotating paratropic/diatropic/paratropic circulations. Graph-theoretical methods have been used to highlight the peculiar topology of these altan-molecules.

Advances of Clar's Aromatic Sextet Theory and Randic´'s Conjugated Circuit Model

Advances of Clar's Aromatic Sextet Theory and Randic´'s Conjugated Circuit Model

The “Anthracene Problem”: Closed-Form Conjugated-Circuit Models of Ring Currents in Linear Polyacenes

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.

Aromaticity and Diatropicity of Polyacenequinododimethides

Typical polyacenequinododimethides exist only in a single classical structure. These hydrocarbons are moderately aromatic and diatropic, although they have no aromatic conjugated circuits. This apparent dichotomy was resolved with our graph theory of aromaticity and magnetotropicity. Many nonconjugated circuits were found to contribute collectively to aromaticity and diatropicity. For individual molecules, local aromaticity increases with distance from the exo-methylene groups. This fact indicates that the conjugated-circuit model is not always applicable to semibenzenoid hydrocarbons such as polyacenequinododimethides.

Aromaticity and magnetotropicity of dicyclopenta-fused polyacenes

Most dicyclopenta-fused polyacenes are predicted to be moderately aromatic and diatropic, although they have no (4n + 2)-site conjugated circuits. We applied our graph theory of aromaticity and magnetotropicity to these molecules and found that these anomalous properties arise from a set of non-conjugated circuits, which contribute collectively to aromaticity and diatropicity. This result indicates that the conjugated circuit model is not always applicable to such non-alternant hydrocarbons. Dianions of dicyclopenta-fused polyacenes are more aromatic than their respective neutral species, because they are iso-pi-electronic with aromatic polyacenes.

An ab Initio Valence Bond Study on Cyclopenta-Fused Naphthalenes and Fluoranthenes

To probe the effect of external cyclopenta-fusion on a naphthalene core, ab initio valence bond (VB) calculations have been performed, using strictly atomic benzene p-orbitals and p-orbitals that are allowed to delocalize, on naphthalene (1), acenaphthylene (2), pyracylene (3), cyclopenta[b,c]acenaphthylene (4), fluoranthene (5), and cyclopenta[c,d]fluoranthene (6). For the related compounds 1-4 and 5,6 the total resonance energies (according to Pauling's definition) are similar. Partitioning of the total resonance energy in contributions from the possible 4n + 2 and 4n pi-electron conjugated circuits shows that only the 6pi-electron conjugated circuits (benzene-like) contribute to the resonance energy. The results show that cyclopenta-fusion does not extend the pi system in the ground state; the five-membered rings act as peri-substituents. As a consequence, the differences in (total) resonance energy do not coincide with the differences in thermodynamic stability. Notwithstanding, the relative energies of the Kekule structures can be estimated using Randic's conjugated circuits model.

Recursive Formulae for Enumeration of LM-Conjugated Circuits in Structurally Related Benzenoid Hydrocarbons

The linearly independent and minimal conjugated (LM-conjugated) circuits of benzenoid hydrocarbons play the central role in the conjugated circuit model. For a general case, the enumeration of LM-conjugated circuits may be tedious as it requires construction of all Kekule structures. In our previous work, a recursive method for enumeration of LM-conjugated circuits of benzenoid hydrocarbons was established. In this paper, we further extend the recursive formulae for enumerations of LM-conjugated circuits for both catacondensed benzenoid hydrocarbons and some families of structurally related pericondensed benzenoid hydrocarbons.

Triscyclopenta[cd,fg,jk]pyrene: another congener of the externally cyclopenta-fused pyrene series

Flash vacuum thermolysis of 1,3,6-tris(1-chloroethenyl)pyrene 5 between 900–950 °C gave the thermally labile triscyclopenta[cd,fg,jk]pyrene 1; in line with the ‘conjugated circuits model’ and supported by IGLO III//RHF/6-31G calculations, all hydrogen atoms of 1 are additionally shielded by 0.4–0.9 ppm (1H NMR).

Stability of fullerenes with four-membered rings

Fullerenes with four-membered rings with up to 60 atoms were systematically generated and their stability examined by the conjugated circuits (CC) model and the topological resonance energy (TRE) model. As none of the models accounts for strain, which is important in spherically shaped conjugated systems with four-membered rings, the results are only qualitative. Values of resonance energy obtained by the two models show remarkable disagreement. Isomers selected as the most stable are quite different in dependence on the criterion used. The structures selected by TRE possess a small number or no four-membered rings (for C 60, buckminsterfullerene was selected), while those selected by the CC model sometimes possess greater numbers (up to 6) of four-membered rings, paired by a common edge.

Matching polynomial and topological resonance energy of C 70

Matching polynomial and its zeros for the molecular graph of C 70 are reported. Topological resonance energy (TRE) is calculated and compared with that reported for C 60, and with the values obtained by conjugated circuits model. TRE values indicate slightly greater stability of C 70, in agreement with recent experimental results.

Is ?4n + 2? a general quantum electromagnetic topological invariant?

We begin with a review of past work using a “gauge model” to compare the phase (or gauge) similarities of Hückel's and the Woodward–Hoffmann rules, and the Aharonov–Bohm effect. The conjugated circuits model provides a clearer description of the connection of aromaticity with the band model than previously used. A common attribute is the effect of a circular path enclosing at least one singularity which creates a nonsimply connected manifold in the presence of a vector potential, Ā. This condition leads to Dirac's ambiguity in the resultant magnetic field . A solution is a Dirac-like monopole proposed by Wu and Yang obtained by coordinate patching around the singularity. Another model attribute is the conservation of angular momentum of the molecule plus field. This obtains by consideration of the return flux, which links the circle of atoms in the molecule with a circle of flux, and provides a “linking” of the two circles. The linking is described by one of the oldest topological invariants, the “Gausslinking integral.” By expanding the monopole solution we can describe the linking integral by means of the (S3 → S2) Hopf map, which necessitates adding a Chern–Simons term to describe this effect properly. Following a brief description of the Chern–Simons basis for the Jones-Witten topological knot theory, we conclude that there are three possible factors which could be responsible for the WH/Hückel 4n + 2 effect: curvature, torsion, and writhing. In this model the monopole (curvature) accounts for the 2, the torsion (orbital) effect for 4n, and the writhing (spin) for 0. Because a topological theory has no metric, it has no size dependence; hence, the model will support a “shell structure” of the periodic table based on 4n + 2. We close with a discussion of the integer quantum Hall effect (IQHE), where the commutivity of translation operators is combined with gauge transformations, thereby defining magnetic translation operators. The same selection rule for commutivity of the magnetic translation operators in the IQHE seems to apply in 4n and 4n + 2 ring compounds. © 1995 John Wiley & Sons, Inc.

Conjugated-circuit computations on two-dimensional carbon networks

First, a new approach to treat the strongly correlated conjugated-circuit model on two-dimensional networks is made with computational effort comparable to that for corresponding tight-binding models. Toward this end a translationally symmetric arrow assignment is used to construct an antisymmetrically signed ‘‘adjacency’’ matrix for two-dimensional networks. Then symmetry blocking is used to manipulate this ‘‘adjacency’’ matrix, and make the associated conjugated-circuit computations. Second, a series of two-dimensional translationally symmetric structures related to graphite is constructed by means of a kind of local rearrangement on the graphite lattice. A consequent detailed description of π-electron resonance energy via conjugated-circuit computations is presented for these novel two-dimensional nets as well as several other regular and semiregular nets with vertices of degree 3. Approximate energy estimates indicate that resonance stability depends dominantly on the local topology of the networks, and in particular on the fraction of faces which are hexagonal.

On conjugated-circuit polynomials

Recursive formulas are derived for generating the conjugated-circuit polynomials for circuits up to size n = 6, 10, or 14 on regular chain polyhex graphs. These counting polynomials are of use for a simple valence-bond resonance-theoretic derived model: the conjugated-circuit model. © 1994 John Wiley & Sons, Inc.

Topological resonance energy predictions of the stability of fullerene clusters

The topological resonance energies (TRE) for various fullerene clusters (C 20 to C 60) have been computed and based on this, stabilities of fullerenes are discussed. Trend in stability predicted by the TRE model is compared with predictions from the conjugated circuit model as well as the MNDO and ab initio methods. Reasonable agreement is found between the TRE model and rigorous quantum mechanical methods.