Acyclic and Characteristic Polynomial of Regular Conjugated Polymers and Their Derivatives

A method to study the acyclic and characteristic polynomial of regular conjugated polymers is described. For a regular polymer with l bonds linking the monomer units, one first builds a 2\u27X21 polynomial matrix T1. Its matrix elements are acyclic polynomials of the monomer unit graph and its subgraphs obtained by successive deletion of atoms serving as the linking sites. The acyclic polynomials of the fasciagraph (representing an open polymeric chain) and some of its subgraphs are then obtained as the appropriate matrix elements of Ti" where M stands for the degree of polymerization of the polymer under consideration. For the rotagraph (representing the polymeric chain closed on itself) the acyclic polynomial equal the trace of T1". It is proved that the acyclic polynomials of regular polymers and some of their derivatives satisfy recursion formulae of the same form which contain 21 + 1 terms. The coefficients appearing in the recursion are derived only from the knowledge of the matrix Ti and are, therefore, independent of M. As far as the characteristic polynomial of a regular polymer is concerned, here we apply an analogon of the Ti-formalism only for the special case of l = 1 and reproduce an already known recursion formula. However, a new determinantal representation of the characteristic polynomial of a polymer as well as its explicit expression in terms of the characteristic polynomials of monomer graph and its subgraphs is established for this special case

Acyclic and Characteristic Polynomial of Regular Conjugated Polymers and Their Derivatives

A method to study the acyclic and characteristic polynomial of regular conjugated polymers is described. For a regular polymer with l bonds linking the monomer units, one first builds a 2\u27X21 polynomial matrix T1. Its matrix elements are acyclic polynomials of the monomer unit graph and its subgraphs obtained by successive deletion of atoms serving as the linking sites. The acyclic polynomials of the fasciagraph (representing an open polymeric chain) and some of its subgraphs are then obtained as the appropriate matrix elements of Ti" where M stands for the degree of polymerization of the polymer under consideration. For the rotagraph (representing the polymeric chain closed on itself) the acyclic polynomial equal the trace of T1". It is proved that the acyclic polynomials of regular polymers and some of their derivatives satisfy recursion formulae of the same form which contain 21 + 1 terms. The coefficients appearing in the recursion are derived only from the knowledge of the matrix Ti and are, therefore, independent of M. As far as the characteristic polynomial of a regular polymer is concerned, here we apply an analogon of the Ti-formalism only for the special case of l = 1 and reproduce an already known recursion formula. However, a new determinantal representation of the characteristic polynomial of a polymer as well as its explicit expression in terms of the characteristic polynomials of monomer graph and its subgraphs is established for this special case

The Effect of Varying Short-Chain Alkyl Substitution on the Molar Absorptivity and Quantum Yield of Cyanine Dyes

The effect of varying short-chain alkyl substitution of the indole nitrogens on the spectroscopic properties of cyanine dyes was examined. Molar absorptivities and fluorescence quantum yields were determined for a set of pentamethine dyes and a set of heptamethine dyes for which the substitution of the indole nitrogen was varied. For both sets of dyes, increasing alkyl chain length resulted in no significant change in quantum yield or molar absorptivity. These results may be useful in designing new cyanine dyes for analytical applications and predicting their spectroscopic properties

Optical properties and charge-transfer excitations in edge-functionalized all-graphene nanojunctions

We investigate the optical properties of edge-functionalized graphene nanosystems, focusing on the formation of junctions and charge transfer excitons. We consider a class of graphene structures which combine the main electronic features of graphene with the wide tunability of large polycyclic aromatic hydrocarbons. By investigating prototypical ribbon-like systems, we show that, upon convenient choice of functional groups, low energy excitations with remarkable charge transfer character and large oscillator strength are obtained. These properties can be further modulated through an appropriate width variation, thus spanning a wide range in the low-energy region of the UV-Vis spectra. Our results are relevant in view of designing all-graphene optoelectronic nanodevices, which take advantage of the versatility of molecular functionalization, together with the stability and the electronic properties of graphene nanostructures.Comment: J. Phys. Chem. Lett. (2011), in pres