The evaluation of spectral moments for molecular graphs of phenylenes

The evaluation of some spectral moments for phenylenes is considered. Explicit topological formulae for the fourth, sixth and eighth moments are derived. This was achieved by adopting the method of Hall [1], which was originally applied to benzenoid systems, and applying it to molecular graphs of phenylenes. It is shown that the moments considered in this paper can be expressed in terms of three mutually independent graphical invariants. Received: 17 October 1996 / Accepted: 18 April 1997     

Spectral moments of polymer graphs

Summary It is shown that for anyk, thekth spectral moment of a polymer graph composed ofn monomer units is an exact linear function of the parametern. This linear relation holds for all values ofn, greater than a critical value ₀=₀(k).on leave from Faculty of Science, University of Kragujevac     

Spectral moments of solvated electrons

A moment theory analysis is performed on the optical absorption spectra of surplus electrons localized in various media. The deduced attributes of the excess particle are compared with those obtained by a perturbation treatment of the exact ground state solution of a currently popular model potential for the species. Several frequency dependent observables are evaluated and a source of inadequacy in the current theory, the presence of a Coulomb tail, is revealed.     

The PI index of phenylenes

The Padmakar–Ivan (PI) index is a graph invariant defined as the summation of the sums of n _eu (e|G) and n _ev (e|G) over all the edges e = uv of a connected graph G, i.e., PI(G) = ∑ _e∈E(G)[n _eu (e|G) + n _ev (e|G)], where n _eu (e|G) is the number of edges of G lying closer to u than to v and n _ev (e|G) is the number of edges of G lying closer to v than to u. An efficient formula for calculating the PI index of phenylenes is given, and a simple relation is established between the PI index of a phenylene and of the corresponding hexagonal squeeze.     

Enumeration of the isomers of phenylenes

Summary The [h]phenylene C_6h H_2h+4 isomers are enumerated up toh=12. The numbers are compared with old and new data for C _n H₅ isomers of benzenoids, fluoranthenoids and biphenylenoids.

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Anzahl möglicher Isomerer von Phenylenen

Zusammenfassung Die Anzahl der [h]Phenylen-Isomeren C_6h H_2h+4 wurde bish=12 ausgewertet. Die Zahlen wurden mit alten und neuen Daten für C _n H _s -Isomere von Benzenoiden, Fluoranthenoiden und Biphenyloiden verglichen.

     

Spectral moments of polycyclic aromatic hydrocarbons. Solution of a kinetic problem

The relationship between the rate of supercritical fluid extraction of polycyclic aromatic hydrocarbons (PAHs) from coal-tar pitch and some topological invariants are examined. The aim is to explain the appearance of a minimum value on the activation energy/molar mass curve of the PAHs. Rate constants are approximated using spectral moment expansion. It is shown that the size of the PAHs determines more than 95% of the extraction rate. Activation energy for the extraction of the PAHs was found to increase with increasing molar mass. The appearance of a minimum value on the activation energy/molar mass curve is the consequence of experimental difficulties resulting from the tendency of lower members of the PAHs to sublime.     

A Kekulé structure basis for phenylenes

By means of Kekulé structures and, in particular, their count K, many properties of benzenoid molecules can be rationalized. The analogous properties of non-benzenoid polycyclic conjugated molecules require the consideration of some, but not all, Kekulé structures, whose number is the precisely defined and long time known ‘algebraic structure count’ ASC. In the general case it is not known how to construct a Kekulé structure basis for non-benzenoid molecules, consisting of ASC Kekulé structures, ASC≤K. We now offer a solution of this problem for phenylenes and point out some applications.     

A theoretical study of large planar [N]phenylenes

Ab initio studies of 14 [N]phenylenes containing 12-membered rings furnish geometries, energies, standard heats of formation, NICS(1) values, and proton chemical shifts. The extent of double-bond localization, DeltaR (in A), for each type of the 58 unique six-membered rings-angular, branched, and terminal-varies linearly with both their NICS(1) values and proton resonances. Values of these parameters depend upon the number and type of neighboring rings. DeltaRtot, the molecular sum of the bond localizations, correlates linearly with the ab initio energy and heat of formation. Reactions that conserve the numbers of angular, branched, terminal, and 12-membered rings are thermoneutral, and their net DeltaRtot and DeltaNICS(1) are also nearly zero.     

Cyclotris(paraquat‐p‐phenylenes)

Reported here is the synthesis, solid‐state characterization, and redox properties of new triangular, threefold symmetric, viologen‐containing macrocycles. Cyclotris(paraquat‐p‐phenylene) ( CTPQT⁶⁺ ) and cyclotris(paraquat‐p‐1,4‐dimethoxyphenylene) ( MCTPQT⁶⁺ ) were prepared and their X‐ray single‐crystal (super)structures reveal intricate three‐dimensional packing. MCTPQT⁶⁺ results in nanometer‐sized channels, in contrast with its parent counterpart CTPQT⁶⁺ which crystallizes as a couple of polymorphs in the form of intercalated assemblies. In the solid state, MCTPQT^3(.+) exhibits stacks between the 1,4‐dimethoxyphenylene and bipyridinium radical cations, providing new opportunities for the manipulation and control of the recognition motif associated with viologen radical cations. These redox‐active cyclophanes demonstrate that geometry‐matching and weak intermolecular interactions are of paramount importance in dictating the formation of their intricate solid‐state superstructures.     

Second order Randić index of phenylenes and their corresponding hexagonal squeezes

Given PH a phenylene, and HS its corresponding hexagonal squeeze, their second-order Randić indices are denoted by ²χ(P H) and ²χ(H S), respectively. The expressions of both ²χ(P H) and ²χ(H S) in terms of their inlet features are found, and a simple relation is estabblished between the second order Randić index of a phenylene and of the corresponding hexagonal squeeze.     

Spectral density distribution moments of a Hamiltonian of N coupled Morse oscillators: The large‐N limit

The expressions for moments of spectral density distribution of a Hamiltonian representing a system of N coupled Morse oscillators are reduced to the large‐N limit. The effect of the values of coupling constants on the spectral density distribution is analyzed. For the values of coupling constants within the pertinent range for real molecules, the relations between moments appear to be close to those for a Gaussian distribution. Outside this range, the deviations from the normal distribution are substantial. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Reactions of bis(tetrazole)phenylenes. Surprising formation of vinyl compounds from alkyl halides

The reactions of 1,2-bis(tetrazol-5-yl)benzene (1), 1,3-bis(tetrazol-5-yl)benzene (2), 1,4-bis(tetrazol-5-yl)benzene (3), 1,2-(Bu₃SnN₄C)₂C₆H₄ (4), 1,3-(Bu₃SnN₄C)₂C₆H₄ (5) and 1,4-(Bu₃SnN₄C)₂C₆H₄ (6) with 1,2-dibromoethane were carried out by two different methods in order to synthesise pendant alkyl halide derivatives of the parent bis-tetrazoles. This lead to the formation of several alkyl halide derivatives, substituted at either N1 or N2 on the tetrazole ring, as well as the surprising formation of several vinyl derivatives. The crystal structures of both 1,2-[(2-vinyl)tetrazol-5-yl)]benzene (1-N,2-N′) (1b) and 1,3-bis[(2-bromoethyl)tetrazol-5-yl]benzene (2-N,2-N′) (5d) are discussed.     

On the representation of molecular quadrupole moments in terms of atomic moments

The magnitude and algebraic sign of the molecular quadrupole moments of the homonuclear diatomic molecules N2, O2, F2, P2, S2 and Cl2 are analyzed by expressing them as a sum of the quadrupole moments of the free atoms and an induced molecular quadrupole due to bond formation. This induced molecular quadrupole is further analyzed in terms of in situ atomic dipole and quadrupole moments constructed following the electron partitioning method suggested by Hirshfeld. These in situ moments are interpreted in terms of the sigma and pi character of the chemical bonds and are compared with those predicted by the DMA method of Stone (The Theory of Intermolecular Forces; Clarendon: Oxford, 1996).     

Dipole moments of halogenonitroalkanes

The empirical dependences of the values of the dipole moments of different types of halogenonitroalkanes on the o^*, _R ⁰ constants and the electronegativity of the halogens and on the ^* constants of the hydrocarbon radicals were established.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 3, pp. 626–628, March, 1991.     

Fractional moments and moments of time in boundary diffusion of random coil polymers

Equations for the evaluation of experiments in free diffusion of random coil polymers in dilute solution are derived. Moments of the concentration function with respect to the cell co-ordinate of whole or fractional order are employed. Weight-average diffusion coefficients raised to any positive power and hence negative moments of the molecular weight distribution can be obtained. Among the latter is M_n. Theta conditions are not required. For a narrow-distribution polymer, the linear concentration effect is eliminated in a simple way without additional experiments. Corresponding equations referring to moments with time as the variable of integration are deduced. These should be applied to experiments in bounded diffusion. The statistical weight in the averages is the reciprocal of the weight given by using moments of the cell co-ordinate. Thus the series of weight averages of diffusion coefficients raised to a negative power and positive moments of the molecular weight distribution are obtainable.