Postscript on Viable Ground-States for Calculating Topological π-Electron Ring-Currents Using the Hückel–London–Pople–McWeeny Model

Attention is drawn to the idea that, in the context of the Hückel–London–Pople–McWeeny (HLPM) approach to π-electron ring-currents, the basic Aufbau process can be mimicked by means of a graph-theoretical algorithm and that the outcome is determined solely by the order of the eigenvalues of the arbitrary molecular-graph representing an extant or hypothetical conjugated system. The Aufbau process usually results in a closed-shell ground-state, but sometimes a unique triplet ground-state arises, sometimes doublets, as well as unique ground-states of higher multiplicity, are encountered, and, on occasions, no uniquely defined π-electronic ground-state is established at all. Previously, the only examples of the latter (‘pathological’) case — which, as with triplet ground-states and other ground-states that are not singlets, precludes the possibility of any HLPM calculation — were graphs that are unlikely candidates for being extant or viable conjugated systems. In this note, however, an example is documented of what is, ostensibly, a plausible unsaturated structure — namely, (Coronene)6–. In the conclusion, attention is drawn to a procedure that averages electron occupation amongst the several orbitals of a degenerate shell. This work is licensed under a Creative Commons Attribution 4.0 International License

Some Observations on Triplet Ground-States in the Context of ‘Topological’ (HLPM) Ring-Currents in Conjugated Systems

When the quasi graph-theoretical Hückel–London–Pople–McWeeny (HLPM) approach is used to calculate ‘topological’ π-electron ring-currents and bond-currents in conjugated hydrocarbons, a problem is identified that occurs whenever application of the Aufbau process gives rise to a π-electronic ground-state configuration that is a triplet. This circumstance seems to occur only occasionally and, even when it does, the generally somewhat outré molecular graphs in question appear unlikely to represent extant or viable conjugated systems. The molecular graphs of four examples are used to illustrate this ‘triplet ground-state problem’, only one of which represents a hydrocarbon that has actually been synthesised. It is pointed out that the ‘triplet ground-state problem’ does constitute an intrinsic limitation of the HLPM approach. It is, though, a limitation that is also necessarily inherent in other equivalent (though ostensibly different) methods of calculating magnetic properties due to π-electron ring-currents — methods that are likewise founded on the Hückel molecular-orbital conventions. When a triplet ground-state arises, topological ring-currents and bond-currents cannot be calculated by the HLPM method and its equivalents. Infinite paramagnetism is formally to be predicted in such situations. This work is licensed under a Creative Commons Attribution 4.0 International License

Topological Ring Currents and Bond Currents in Some Neutral and Anionic Altans and Iterated Altans of Corannulene and Coronene.

The novel series of conjugated systems called altans, defined nearly a decade ago, was subsequently extended to multiple ("iterated") altans, and their magnetic properties were calculated by Monaco and Zanasi using the ab initio ipso-centric formalism. Such properties of the single ("mono") altans of corannulene and coronene, calculated by this sophisticated ab initio approach, had earlier been compared with those calculated via the rudimentary Hückel-London-Pople-McWeeny (HLPM) method-a parameter-free topological, quasi graph-theoretical approach requiring knowledge only of the conjugated system's molecular graph and the areas of its constituent rings. These investigations are here extended to double and triple altans. HLPM bond currents in several neutral mono altans are found to differ from those in the corresponding dianion only in those bonds that lie on the structures' perimeters, while the HLPM bond currents in all bonds in the neutral double and triple altans of corannulene and coronene are precisely the same as in the respective dianions. Some rationalization of these unexpected phenomena is offered in terms of the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) nature of the role played by the lone nonbonding orbital in each of the neutral species and its respective dianion

Ring-Current Assessment of the Annulene-Within-an-Annulene Model for some Large Coupled Super-Ring Conjugated-Systems

The Annulene-Within-an-Annulene (AWA) model for conjugated super-ring systems, proposed by Barth and Lawton nearly fifty years ago, is further tested on six newly considered ‘coupled’ structures by means of π-electron ring-currents and bond-currents calculated via the Hückel–London–Pople–McWeeny (HLPM) formalism. Super-ring systems are said to be decoupled when the bonds connecting the central ring to the outer perimeter never appear as anything other than single bonds in any Kekulé structure that can be devised for the system as a whole. The preliminary conclusions of other recent inves¬tigations — by the same HLPM method and by ipso-centric ab-initio approaches — are verified. Phenyl-ene-[5]-circulene, a (coupled) alternant isomer of the much-studied (decoupled) non-alternant system [10,5]-coronene, is also considered. It is advised that the AWA model should in future either be used with consider¬able caution in very specific circumstances, or it should be abandoned altogether

Correlations between Topological Ring-Currents, π-Electron Partitions, and Six-Centre Delocalisation Indices in Benzenoid Hydrocarbons

Comparison is made between three different indices that characterise the individual rings of a wide range of condensed benzenoid hydrocarbons. Two of these ("π-electron partitions" and the six-centre delocalisation-indices that have been called "Δ6-values") have been introduced only recently as potential indicators of what might be called "local aromaticity", whilst the third ("topological π-electron ring-currents") was suggested as a possible discriminator in this regard nearly fifty years ago. Whilst linear correlations between ring currents and π-electron partitions within certain restricted classes of ring types are good (with correlation coefficients of up to 0.998), agreement between the two indices over all classes of ring types is poor. Predictions arising from a consideration of π-electron partitions and Δ6-values seem, on the other hand, to be in somewhat better accord. It is therefore concluded that, despite its superficially intuitive appeal, the ring-current index is out of step both with π-electron partitions and Δ6-values as a general indicator of so-called "local aromaticity". (doi: 10.5562/cca1846

Molecular Complexity of Certain Homologous Series of Condensed Benzenoid Hydrocarbons: Limiting Values of the Patency Index and the Index of Spanning-Tree Density

Two indices of molecular complexity, the Patency Index (2017) and the Spanning-Tree Density (2003), are applied to three homologous series of condensed benzenoid hydrocarbons. Calculation of the Spanning-Tree Density requires finding the number of spanning trees in a given molecular graph, which may be achieved by applying the Cycle Theorem (2004) or, in the case of planar graphs in a planar embedding, the theorem of Gutman, Mallion & Essam (1983). To compute the Patency Index, it is necessary to count the number of ladders in the embedded molecular graph. This is done by means of the Dual Cycle Theorem (2017). In the latter, a ladder is conceived of as the edge-set relating to faces as the edge-set of a spanning tree relates to vertices. For a planar graph in a planar embedding, the number of ladders is equal to the number of spanning trees. The three homologous series investigated here are the Linear [n]-Acenes (An), the [n]-Phenacenes (Phn) and the [n]-Helicenes (Hn) (the latter of which are geometrically non-planar but graph-theoretically planar). For these three series, the Spanning-Tree Densities and the Patency Indices may be obtained in closed form and so their behaviour as n → ∞ is easily examined. Though neither index distinguishes between individual members of the three series (An, Phn and Hn) for a specified value of n, this does not preclude the possibility that, within each series, either index may exhibit correlation with physical or chemical data. This work is licensed under a Creative Commons Attribution 4.0 International License

A Dual of the Cycle Theorem and its Application to Molecular Complexity

The duality alluded to in the title is that between the faces and vertices of a graph embedded on a surface. Its recognition in the context of the five Platonic solids is classic. Algebraically, it is present in the equation for Euler’s Polyhedron Theorem and in the various extensions thereof. The Cycle Theorem (CT) establishes a formula for the number of spanning trees contained in a graph embedded on a surface. It is based on the mutual incidences of its cycles (circuits which also carry a sense of direction), i.e., of sub-graphs of the Cn type, endorsed with a sense. These appear (though not exclusively) as the boundaries of faces, so that, so to speak, the Cycle Theorem establishes a result which is essentially about vertices via relations between faces. Among several possible duals of the Cycle Theorem there might thus be one that estab¬lishes a relation which is essentially about faces via relations between vertices. In order to formulate one, we define, for an embedded graph, a feature concerning faces that is dual to a spanning tree. We call it a ladder. A formula is presented for the number of ladders contained in a graph which, in some cases, introduces the concept of ‘artificial vertices’. It is based on the mutual incidences of its vertices. Its form is clearly analogous, or ‘dual’, to the Cycle Theorem formula for spanning trees, previously proposed (in this journal — 2004) by three of the present authors, together with Klein and Sachs. A new index is proposed, which involves ladders. We call it the Patency Index of a graph; its numerical value may be related to molecular complexity. It is effectively the dual of, and is entirely analogous to, the Spanning-Tree Density Index which was earlier (2003) proposed, defined and applied to molecular graphs by one of the present authors and Trinajstić. This work is licensed under a Creative Commons Attribution 4.0 International License

What Kirchhoff Actually did Concerning Spanning Trees in Electrical Networks and its Relationship to Modern Graph-Theoretical Work

What Kirchhoff actually did concerning spanning trees in the course of his classic paper in the 1847 Annalen der Physik und Chemie has, to some extent, long been shrouded in myth in the literature of Graph Theory and Mathematical Chemistry. In this review, Kirchhoff’s manipulation of the equations that arise from application of his two celebrated Laws of electrical circuits — formulated in the middle of the 19th century — is related to 20th- and 21st-century work on the enumeration of spanning trees. It is shown that matrices encountered in an analysis of what Kirchhoff really did include (a) the Kirchhoff (Laplacian, Admittance) matrix, K, that features in the well-known Matrix Tree Theorem, (b) the matrix G encountered in the theorem of Gutman, Mallion & Essam (1983), applicable only to planar graphs, and (c) the analogous matrix M that arises in the Cycle Theorem (Kirby et al. 2004), a theorem that applies to graphs of any genus. It is concluded that Kirchhoff himself was not interested in counting spanning trees, and, accordingly, he did not explicitly do so. Nevertheless, it is shown how the modulus of the determinant of a certain matrix (here denoted by the label C\u27) — associated with the linear equations arising from application of Kirchhoff’s two Laws — is numerically equal to the number of spanning trees in the graph representing the connectivity of the electrical network being studied. Kirchhoff did, however, invoke the concept of spanning trees, introducing them in a complementary fashion by referring to the chords that must be removed from the original graph in order to form such trees. It is further emphasised that, in choosing the cycles in the network being studied, around which to apply his circuit Law, Kirchhoff explicitly selected what would now be called a ‘Fundamental System of Cycles’. This work is licensed under a Creative Commons Attribution 4.0 International License

A Theorem for Counting Spanning Trees in General Chemical Graphs and Its Particular Application to Toroidal Fullerenes

A theorem is stated that enables the number of spanning trees in any finite connected graph to be calculated from two determinants that are easily obtainable from its cycles → edges incidence-matrix. The 1983 theorem of Gutman, Mallion and Essam (GME), applicable only to planar graphs, arises as a special case of what we are calling the Cycle Theorem (CT). The determinants encountered in CT are the same size as those arising in GME when planar graphs are under consideration, but CT is applicable to non-planar graphs as well. CT thus extends the conceptual and computational advantages of GME to graphs of any genus. This is especially of value as toroidal polyhexes and other carbon-atom species embedded on the torus, as well as on other non-planar surfaces, are presently of increasing interest. The Cycle Theorem is applied to certain classic, and other, graphs – planar and non-planar – including a typical toroidal polyhex