Real space electron delocalization, resonance, and aromaticity in chemistry

Chemists explaining a molecule’s stability and reactivity often refer to the concepts of delocalization, resonance, and aromaticity. Resonance is commonly discussed within valence bond theory as the stabilizing effect of mixing different Lewis structures. Yet, most computational chemists work with delocalized molecular orbitals, which are also usually employed to explain the concept of aromaticity, a ring delocalization in cyclic planar systems which abide certain number rules. However, all three concepts lack a real space definition, that is not reliant on orbitals or specific wave function expansions. Here, we outline a redefinition from first principles: delocalization means that likely electron arrangements are connected via paths of high probability density in the many-electron real space. In this picture, resonance is the consideration of additional electron arrangements, which offer alternative paths. Most notably, the famous 4n + 2 Hückel rule is generalized and derived from nothing but the antisymmetry of fermionic wave functions.

Benzene-like N6rings in a Be2N6monolayer: a stable 2D semiconductor with high carrier mobility

Abundant delocalized six-centre–two-electron (6c–2e) π-bonds are responsible for the planar configuration and high structural stability of a Be2N6monolayer, which satisfies the Hückel rule [4n+ 2] (n= 2).

Application of graph theory and topological models for the determination of fundamentals of the aromatic character of pi-conjugated hydrocarbons

Application of topological analysis and graph theory to benzenoid hydrocarbons leads to the determination of fundamentals of aromaticity: the Hückel rule and the Clar rule. The approach, based on a treatment of the adjacency matrix, allows resonance energy (RE)-like characteristics to be estimated with quite good accuracy, and magnetic aromaticity indices to be derived for both the individual rings and the whole molecules. It also allows an effective approach for interpreting ring current formation in molecules when exposed to an external magnetic field. The transformation of the perturbation matrix into a form describing the canonical structures allows their gradation and determination of their stabilizing/destabilizing character.

Variation of aromaticity by twisting or expanding the ring content

Generalization of the Hückel rule predicts that the (anti)aromaticity of a neutral ring is qualitatively reverted upon a single twist of the π-orbital array (Möbius interconversion), and is preserved upon expansion of all the bonds by single C2 units (ring carbo-merization). These opposite effects are addressed from quantitative theoretical and experimental standpoints, respectively. (i) According to most resonance energy (RE) schemes, the RE value of a Möbius ring is not the opposite of that of the Hückel version. This also applies to the Aihara’s and Trinajstic’s topological resonance energy (TRE), where a non-aromatic reference in the topological limit is defined as being “as identical as possible” to the parent ring but just “acyclic”. In spite of its conceptual merits, the computing complexity and fictitious character of the TRE acyclic reference resulted in a disuse of TRE as a current energetic aromaticity index. Both the calculation and interpretation of TRE have been revisited in light of a cross-reference between the Hückel and Möbius rings within the Hückel molecular orbital (HMO) framework. Whereas the topological influence of triple bonds is currently neglected in the first-level HMO treatment of π-conjugated systems, a graph-theoretical analysis allows one to differentiate the TRE value of a [3n]annulene from those of the corresponding carbo-[n]annulene. The C18 ring of carbo-benzene is thus predicted to be slightly more topologically aromatic than that of [18]annulene. (ii) Recent experimental and density functional theory (DFT) theoretical studies of quadrupolar carbo-benzene derivatives are presented. The results show that the “flexible aromaticity” of the p-C18Ph4 bridge between donor anisyl substituents plays a crucial role in determining the intriguing chemical/spectroscopical/optical properties of these carbo-chromophores.

Theory of moments, Hückel rule and stability

The connection between moments of the electronic Hamiltonian and topology of a quantum mechanical system is studied. Based on simplifications similar to those usually employed in simple chemical and physical theories, criteria resembling the Hückel rule for cyclic conjugated systems are suggested. Several examples of interest in chemistry and solid physics are discussed. No information on the wave function is necessary in the present approach.