Isomers of Polyenes Attached to Benzene

A polyene graph is a tree that can be embedded in a hexagonal lattice. Systems of polyene graphs attached to one hexagon are considered. Overlapping edges and/or vertices (geometrically nonplanar systems) are allowed. A complete mathematical solution is presented in terms of a generating function for the numbers of isomers of the systems in question. The corresponding geometrically planar systems, referred to as styrenoids, are enumerated by computer programming. Finally, in the Appendix, the generating function is given for the numbers of free polyene graphs

GRAPH-THEORETICAL STUDIES ON FLUORANTHENOIDS AND FLUORENOIDS - ENUMERATION OF SOME CATACONDENSED SYSTEMS

Precise definitions are given for some classes of molecular graphs with one pentagon and otherwise hexagons: the monopentapolyhexes. The fluoranthenoid and fluorenoid systems belong to monopentapolyhexes. Complete mathematical solutions, using combinatorial summations on the one hand and generating functions on the other hand, are given for the numbers of catacondensed simply connected monopentapolyhexes (catafluorenoids and the corresponding helicenic systems). Generating functions and numerical values are included

Raman scattering in C_{60} and C_{48}N_{12} aza-fullerene: First-principles study

We carry out large scale {\sl ab initio} calculations of Raman scattering activities and Raman-active frequencies (RAFs) in ${\rm C}_{48}{\rm N}_{12}$ aza-fullerene. The results are compared with those of ${\rm C}_{60}$. Twenty-nine non-degenerate polarized and 29 doubly-degenerate unpolarized RAFs are predicted for ${\rm C}_{48}{\rm N}_{12}$. The RAF of the strongest Raman signal in the low- and high-frequency regions and the lowest and highest RAFs for ${\rm C}_{48}{\rm N}_{12}$ are almost the same as those of ${\rm C}_{60}$. The study of ${\rm C}_{60}$ reveals the importance of electron correlations and the choice of basis sets in the {\sl ab initio} calculations. Our best calculated results for ${\rm C}_{60}$ with the B3LYP hybrid density functional theory are in excellent agreement with experiment and demonstrate the desirable efficiency and accuracy of this theory for obtaining quantitative information on the vibrational properties of these molecules.Comment: submitted to Phys.Rev.