Topological Resonance Research Articles

How Does Aromaticity Rule the Thermodynamic Stability of Hydroporphyrins?

Several measures of aromaticity including energetic, magnetic, and electron density criteria are employed to show how aromatic stabilization can explain the stability sequence of hydroporphyrins, ranging from porphin to octahydroporphin, and their preferred hydrogenation paths. The methods employed involve topological resonance energies and their circuit energy effects, bond resonance energies, multicenter delocalization indices, ring current maps, magnetic susceptibilities, and nuclear-independent chemical shifts. To compare the information obtained by the different methods, the results have been put in the same scale by using recently proposed approaches. It is found that all of them provide essentially the same information and lead to similar conclusions. Also, hydrogenation energies along different hydrogenation paths connecting porphin with octahydroporphin have been calculated with density functional theory. It is shown by using the methods mentioned above that the relative stability of different hydroporphyrin isomers and the observed inaccessibility of octahydroporphin both synthetically and in nature can be perfectly rationalized in terms of aromaticity.

On the dielectric function tuning of random metal-dielectric nanocomposites for metamaterial applications

The potential of random metal-dielectric nanocomposites as constituent elements of metamaterial structures is explored. Classical effective medium theories indicate that these composites can provide a tunable negative dielectric function with small absorption losses. However, the tuning potential of real random composites is significantly lower than the one predicted by classical theories, due to the underestimation of the spectral range where topological resonances take place. This result suggests that a random mixture consisting of a metal matrix with embedded isolated dielectric inclusions is a promising design guideline for the fabrication of tunable composites for metamaterial purposes.

ChemInform Abstract: Variation of Aromaticity by Twisting or Expanding the Ring Content

AbstractReview: 105 refs.

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 singleC2units (ringcarbo-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 isnotthe 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 correspondingcarbo-[n]annulene. The C18ring ofcarbo-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 quadrupolarcarbo-benzene derivatives are presented. The results show that the “flexible aromaticity” of thep-C18Ph4bridge between donor anisyl substituents plays a crucial role in determining the intriguing chemical/spectroscopical/optical properties of thesecarbo-chromophores.

The chemical roots of the matching polynomial

The interpretation of topological resonance energy (TRE), the mathematical measure of energetic chemical aromaticity at the Hückel level of theory, is revisited by providing a concise analysis of the matching polynomial of a chemical graph. Whereas the matching (or acyclic) polynomial is not a properly characteristic polynomial in general, it is the arithmetic mean of the characteristic polynomials of the corresponding Hückel and Möbius chemical graphs. For unicyclic graphs, the roots of the matching polynomial (serving to calculate the TRE values) can be graphically approximated from generalized Frost-Musulin-Zimmermann diagrams, the results becoming exact for annulenic monocycles. The matching polynomial is also shown to be a weighted sum of characteristic polynomials of subgraphs: the whole graph itself and the exocyclic graphs obtained by suppression of individual circuits. The performance of these ring twisting and ring masking processes is compared to that of the previously proposed ring opening process. Further prospects for the interpretation and approximation of TRE are discussed.

Theoretical basis for the correlation between energetic, magnetic, and electron density criteria of aromaticity: definition of molecular circuit electric conductance

Using a semiclassical model, a theoretical basis for the correlation between energetic, magnetic, and electron density criteria of aromaticity has been established. The key quantity that links ring current intensity, topological resonance energy, and multicenter electron delocalization is the molecular circuit electric conductance, which is defined as the inverse of the resistance of the molecular circuit to the electron flow when an external electromotive force is exerted on it. The particular case of an external magnetic field applied perpendicularly to the molecular plane is employed here as source of the electromotive force. In addition, it is shown how aromatic stabilization can be estimated quantitatively by means of multicenter electron delocalization in PAHs and nitrogenated polyheterocyclic compounds.

Responsive Hybrid Nanotubes Composed of Block Copolymer and Gold Nanoparticles

We demonstrate the facile fabrication of high aspect ratio nanotubes from responsive block copolymer, polystyrene-block-poly(2-vinylpyridine) (PS-b-P2VP), and in situ grown gold nanoparticles. PS-b-P2VP nanotubes are fabricated by template-assisted assembly by utilizing cylindrical pores of porous alumina membranes followed by its dissolution. Solvent annealing was employed to facilitate ordered phase separation of the block copolymer with in the nanotube structure, followed by in situ formation of gold nanoparticles in the P2VP domains occupying the inner and outer shell of the nanotubes. Owing to the pH-sensitive nature of P2VP block, which exhibits a coil to globule transition above pH 3.6, the PS-b-P2VP−gold nanoparticles hybrid nanotubes show reversible changes in topology, size (up to 25%), and collective plasmon resonance appearance. We suggest that PS-b-P2VP−gold nanoparticles hybrid nanotubes with their pH-responsive properties, large surface-to-volume ratio, and through open pore can find intriguing applications in sensing and microfluidic systems.

New developments in Hückel theory

An aromaticity index, called the topological resonance energy, TRE, which is a nonparametric, Dewar-like resonance energy, is discussed, TRE is defined as the difference between the total Hückel energy calculated for a given conjugated molecule and that calculated from the corresponding acyclic polynomial. The graph-theoretical rules for computing the acyclic polynomial are given. TRE index is free of the limitations present in other resonance energy indices and may be straightforwardly applied to conjugated molecules, their ions and radicals, homoaromatic systems, etc.

Origin of Stacked-Ring Aromaticity

Although strong diatropic currents run in the pi-stacked dimers of 4n pi annulenes, they are still antiaromatic with negative topological resonance energies (TREs). The TRE difference between two annulene molecules and the pi-stacked annulene dimer can be used as a practical measure of stacked-ring aromaticity. As predicted by Corminboeuf et al., all pi-stacked 4n pi annulene dimers exhibit marked stacked-ring aromaticity. In contrast, the TRE for the entire conjugated system of a pi-stacked benzene dimer is much smaller than twice the TRE for benzene. The analysis of circuit contributions to energetic and magnetic properties shows that such stacked-ring aromaticity and diatropicity arise from many (4n + 2)-site circuits created by the stacking of two antiaromatic rings. Tetragonal faces formed in the dimers are responsible for the creation of these (4n + 2)-site circuits.

The close relation between cyclic delocalization, energy effects of cycles and aromaticity

New evidence questioning the multidimensionality of the aromaticity phenomenon exemplified in what is called orthogonality between the classical (structural and energetic) and magnetic aromaticity indices and measures is reported. For this purpose, the recently proposed methodologies for the quantitative characterization of the energy benefits associated with the cyclic arrangement of mobile π-electrons in polycyclic aromatic hydrocarbons are compared with the indices characterizing the extent of cyclic delocalization in the corresponding conjugated circuits. The reported close correlation between both types of indices implies that no discrepancies between classical and magnetic aromaticity measures exist provided the comparison is based on the indices of inherently local nature and/or the interfering contributions of contaminating conjugated circuits is properly taken into account in the description of aromaticity measures like topological resonance energy (TRE) or nucleus independent chemical shift (NICS).

Aromaticity of Heterofullerenes C18BxNy(x+y= 2) and Their Molecular Ions

The aromaticity of all possible substituted fullerene isomers of C18N2, C18B2, C18BN, and their molecular ions which originate from the C20 (Ih) cage were studied by the topological resonance energy (TRE) and the percentage topological resonance energy methods. The relationship between the aromaticity of C18BxNy isomers and the sites where the heteroatoms dope at the C20 (Ih) cage is discussed. Calculation results show that at the neutral and cationic states all the isomers are predicted to be antiaromatic with negative TREs, but their polyvalent anions are predicted to be aromatic with positive TREs. The most stable isomer is formed by heteroatom doping at the 1, 11-sites in C18N2, C18B2, and C18BN. Heterofullerenes are more aromatic than C20. The stability order in the neutral states is C18N2 > C18 BN > C18 B2 > C20. The stability order in closed-shell is C18B28 > C206 > C18BN6 > C18N24. This predicts theoretically that their polyvalent anions have high aromaticity.

Aromaticity of All Possible C26N2 Isomers

AbstractThe aromaticity of all possible heterofullerenes C26N2 and C28 based on Td symmetry has been studied by means of the topological resonance energy and percentage topological resonance energy methods. The relationship between the aromaticity of the C26N2 isomers and the sites where nitrogen atoms dope at the C28 cage has been discussed. The calculation results show that the most stable isomer of C26N2 derivatives is formed by nitrogen atoms doping at the two tetrahedral vertices. C26N2 isomers are more stable than C28, but the C26N22− isomers are less stable than C28 4−4. The effect of nitrogen substitution on C28 stability was investigated by the topological charge stabilization rule.

Topological resonance energy, bond resonance energy, and circuit resonance energy

AbstractCircuit resonance energy (CRE) is a key quantity that links energetic and magnetic criteria of aromaticity for a polycyclic π‐system. Topological resonance energy (TRE) correlates well with the sum of CREs for an entire π‐system. We found that bond resonance energy (BRE), so far defined to estimate the degree of kinetic stability for a polycyclic π‐system, also correlates very well with the corresponding quantity defined with CREs. These findings justify the view that graph‐theoretical quantities, such as TRE and BRE, can be used safely even though they rely on unrealistic reference structures. Copyright © 2007 John Wiley & Sons, Ltd.

Magnetic Resonance Energies of Heterocyclic Conjugated Molecules

Magnetic resonance energy (MRE), derived from ring-current diamagnetic susceptibility, can be interpreted as a kind of aromatic stabilization energy. For polycyclic conjugated hydrocarbons, this quantity correlates well with topological resonance energy (TRE). MREs for typical heterocyclic conjugated molecules were then calculated and analyzed. It was found that even for heterocycles MRE highly correlates with TRE. Thus, the MRE concept has been firmly established as a reliable indicator of aromaticity, which mediates magnetic criteria of aromaticity with energetic ones. The conformity of heterocycles to the rule of topological charge stabilization can be checked using not only TRE but also MRE.

The Cycle‐Dependence of Topological Resonance Energy.

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Properties of Aromaticity Indices Based on the One-Electron Density Matrix

Proper normalization of two previously published indices yields aromaticity measures that, when computed within the Hückel molecular orbital (HMO) approximation, closely match the topological resonance energies per pi electron of aromatic annulenes and their ions. The normalized indices, which quantify aromaticity of individual rings in polycyclic systems, are equally applicable to homocyclic and heterocyclic compounds and can be readily computed from 1-matrices calculated at any level of electronic structure theory. However, only the index ING, derived from the Giambiagi formula, produces proper ordering of aromaticities of heterocyclic compounds, provided it is calculated from all-electron wavefunctions in conjunction with the atoms in molecule (AIM) partitioning. Its values are shown to be strongly affected by electron correlation effects. Because of its apparent inability to distinguish between anti- and nonaromatic systems, ING should only be employed for aromatic species.

The quantum states of high- Tc materials – States of topological resonances

The nature of the electronic states of the CuO 2 planes in high- T c materials is elaborated, being proven as a crucial revision of previous assumptions. One important aspect is the general existence of electronic pair states in two manifestations, which determine superconducting or non-superconducting properties. The decisive basis of the theory developed in this paper is derived from the various broken symmetries of the electronic states and the formation of one-dimensional electronic subspaces. The non-superconducting state is already characterized by a two-dimensional ordering of one-dimensional electronic states in space ( stiffness in space), whereas the superconducting state is represented by a two-dimensional highly correlated electronic state in space and time ( stiffness in space and time). Under the inclusion of lattice dynamics, the electronic ground state is reached by a topological superposition of symmetry-broken sub-states, quantized and determined in time ( Topological Resonance (TR)). This forms the basis for inherent times ( eigentime) of the electronic quantum states, an important but unusual characteristic of these quantum states. Under hole doping, the general nature of the electronic state of the CuO 2 plane can be characterized as a coherent quantum fluid consisting of two partial fluids (b-holes, f-holes). The characteristics of this quantum fluid state are predominantly determined by the strongly correlated b-hole fluid, whereas the f-hole fluid possesses higher internal degrees of freedom. The existence of a b-hole fluid is an indispensable precondition for the existence of a superconducting state in high- T c materials.

On the Cycle-Dependence of Topological Resonance Energy

The topological resonance energy (TRE) was conceived in the 1970s. From the very beginning, it was known that TRE is equal to the collective energy-effect of all cycles present in a conjugated molecule. Also in the 1970s a theory of cyclic conjugation was elaborated, by means of which it was possible to compute the energy-effect ef(Z) of each individual cycle Z present in a conjugated molecule. Yet, the connection between TRE and the ef(Z)-values was, until now, not studied. We now show that TRE and the sum of the ef(Z)-values are closely correlated but that a certain correction needs to be made by taking into account the energy-effects of pairs, triplets, quartets, etc. of cycles.

Topological Translations and Nonlinear Resonance

Abstract In this paper we study resonance for nonlinear oscillators from a topological point of view. The Poincaré map associated to a linear second order equation at resonance is a translation of the plane. Our aim is to prove that in the case of nonlinear equations the Poincaré map is topologically conjugate to a translation. The proof is based on a characterization of translations in terms of Lyapunov functions.

Relating resonance energy with the Zhang-Zhang polynomial

The Zhang-Zhang polynomial ?(x) is a recently conceived tool in the mathematical apparatus of theoretical chemistry. It combines (in a quantitative manner) the Kekul?- and Clar-structure-based features of benzenoid molecules. It is shown that the topological resonance energy (TRE) can be accurately approximated as TRE ? a??(x)+ b where x?0.5, by which fact a significant insight is gained into the structure-dependence of TRE of benzenoid molecules.