Graph theoretical procedure for obtaining analytical expressions of eigenspectra of linear chains and cycles with alternant vertex weights and same edge weight: Application to some complicated graphs

A graph theoretical procedure for obtaining eigenvalues of linear chains and cycles having alternant vertex weights (h₁, h₂, h₁, h₂, h₁, h₂, …) and the same edge weight (k) have been developed. The eigenvalues of some complicated graphs, such as graphs of linear polyacenes, methylene‐substituted linear polyacenes and cylindrical polyacene strips, stack graphs, and reciprocal graphs have been shown to be generated in closed analytical forms by this procedure. Many such graphs represent chemically important molecules or radicals. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Characteristic polynomials of linear polyacenes and their subspectrality

Coefficients of characteristic polynomials (CP) of linear polyacenes (LP) have been shown to be obtainable from Pascal’s triangle by using a graph factorisation and squaring technique. Strong subspectrality existing among the members of the linear polyacene series has been shown from the derivation of the CP’s. Thus it has been shown that the entire eigenspectrum of ann-ring LP is included in that of (2n + 1)-ring LP. Correspondence between the eigenspectra of linear chains and LP’s is brought out by a recently developed vertex-alternation and squaring algorithm.     

Visible-Light-Induced Heptacene Generation under Ambient Conditions: Utilization of Single-crystal Interior as an Isolated Reaction Site

The photo-induced generation of unstable molecules generally requires stringent conditions to prevent oxidation and the concomitant decomposition of the products. The visible-light-induced conversion of two heptacene precursors to heptacene was studied. Single crystals of bis- and mono-α-diketone-type heptacene precursors ( 7-DK2 and 7-DK1 , respectively), were prepared to investigate the effect of precursor structure on reactivity. The photoirradiation of a 7-DK2 single crystal cleaved only one α-diketone group, forming an intermediate bearing a pentacene subunit, while that of a 7-DK1 single crystal gave rise to characteristic absorption peaks of heptacene and their increase in intensity with photoirradiation time, indicating the generation of heptacene without decomposition. Heptacene production was not observed when the precursors were photoirradiated in solution, implying that the single crystal interior provided isolation from the external environment, thus preventing heptacene oxidation.     

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.     

Spectral‐Luminescence Characteristics of Polyphenyls and Polyacenes in the Gas Phase in Electron‐Beam Excitation

The spectra of electronenergy loss, excitation functions, and fluorescence spectra in excitation of the vapor of polyphenyls and polyacenes by electron beams of different energies are determined. The influence of successive complication of the molecules under study on these spectralluminescence characteristics is tracked. Unlike the optical absorption spectra, in the spectra of electronenergy loss of all the substances studied one observes a band which is related to the singlettriplet transition S ₀–T ₁. The transitions up to S ₀–S ₅ are recorded in excitation of the molecules by highenergy electrons, including the region of vacuum ultraviolet. From the functions of fluorescence excitation the authors have determined the excitation thresholds that correlate with the energies of the S ₁ levels, except for pyrene in which the S ₀–S ₁ transition is forbidden and does not show up not only in photon excitation but also in electronbeam excitation, although the intercombination forbiddenness in the latter case is removed and the S ₀–T ₁ band is observed.     

Use of symmetry plane fragmentation and graph squaring techniques to express the eigenspectra for some vertex-weighted graphs of linear chains and cycles in analytical form

Three types of graphs of linear chains, viz. linear chains with unit increment or decrement in weight on one terminal vertex, linear chains with unit increment or decrement in weight on both the terminal vertices and linear chains with unit increment in weight on one terminal vertex and decrement in that on the other terminal vertex, have been considered. The symmetry plane fragmentation and graph squaring techniques have been exploited to express the eigenspectra of such graphs of linear chains in analytical form, and have subsequently been used to express the eigenspectra of graphs of linear chains and cycles with alternant vertex weights. The derived expressions for the eigenspectra have been used to obtain the eigenspectra of linear polyacenes, methylene-substituted linear polyacenes and cylindrical polyacene strips in analytical form.     

Orbital optimisation in the perfect pairing hierarchy: applications to full-valence calculations on linear polyacenes

ABSTRACT

We describe the implementation of orbital optimisation for the models in the perfect pairing hierarchy. Orbital optimisation, which is generally necessary to obtain reliable results, is pursued at perfect pairing (PP) and perfect quadruples (PQ) levels of theory for applications on linear polyacenes, which are believed to exhibit strong correlation in the π space. While local minima and σ-π symmetry breaking solutions were found for PP orbitals, no such problems were encountered for PQ orbitals. The PQ orbitals are used for single-point calculations at PP, PQ and perfect hextuples (PH) levels of theory, both only in the π subspace, as well as in the full σπ valence space. It is numerically demonstrated that the inclusion of single excitations is necessary also when optimised orbitals are used. PH is found to yield good agreement with previously published density matrix renormalisation group data in the π space, capturing over 95% of the correlation energy. Full-valence calculations made possible by our novel, efficient code reveal that strong correlations are weaker when larger basis sets or active spaces are employed than in previous calculations. The largest full-valence PH calculations presented correspond to a (192e,192o) problem.     

Novel PI Indices of Hexagonal Chains

Problems don't hung on trees. Formulation of a good problem is often the most important part of a (theoretical) research. New problems usually arise when you try to solve old problems. Ivan Gutman, June 27, 1995.The Padmakar–Ivan (PI) index of hexagonal chains (i.e., the molecular graphs of unbranched catacondensed benzenoid hydrocarbons) is examined. The index PI is a graph invariant defined as the summation of the sums of edges of n _eu and n _ev over all the edges of connected graph G, where n _eu is the number of edges of G lying closer to u than to v and n _ev is the number of edges of G lying closer to v than to u. An efficient calculation of formula for PI for hexagonal chains are put forward.     

Theoretical study of the valence π → π* excited states of polyacenes: anthracene and naphthacene

The valence π → π ^* excited states of anthracene and naphthacene are studied with multireference perturbation theory with complete active space self-consistent field reference functions. The predicted spectra provide a consistent assignment of all one- and two-photon spectra and T-T spectra of low-lying valence π → π ^* excited states of anthracene and naphthacene. The present theory predicts the valence π → π ^* excitation energies with an accuracy of 0.15 eV for anthracene and of 0.25 eV or better for naphthacene. The excited states of anthracene and naphthacene are compared with those of benzene and naphthalene studied previously. The present calculations predict that, going from anthracene to naphthacene, there is a symmetry reversal of the two lowest singlet state transitions, but not for the triplet, just as indicated by the experimental data. Some general trends of polyacene excited states are discussed based on the calculated results for benzene to naphthacene. Conclusive results obtained for anthracene and naphthacene can be used as a model for understanding the excited states of larger polyacenes. Received: 22 April 1998 / Accepted: 6 July 1998 / Published online: 28 September 1998