A Systematic Study of Symmetry Properties of Graphs I. Petersen Graph

Recently (Chem. Phys. Lett. 42 283 (1976) a simple procedure for deriving symmetry properties of graphs has been suggested. It is based on a canonical numbering of the vertices of a graph, and consists in searching for all the acceptable numberings which have a unique adjacency matrix. In the series of papers initiated here we will apply the above procedure and derive all symmetry operations for graphs of interest to chemistry. We start with the Petersen graph which is of interest in discussions of isomerizations of trigonal bipyramidal structures

Hybridization in Planar XY 4 Molecules

Maximum overlap hybriid orbitals for planar XY4 molecules are described assuming various orbitals for ligand and central atoms. Distribution of s and d character in square snp2d2-n hybridization as a tunction of ilnteratomilc distances, and for different relamive effective charges iof bonded atoms .is g.iven. Maximum overlap hybdds, when compared with those based on Pauling\u27s criterion of hybrid 1strength, show a generally smaller contribuition of d z2 orbdtal in the hybnidizati\u27on, especially evident for ~ligand< μm etal, when in SOlffie cases they _ reduce to simple dsp2 hybrids . Calculait:i.ons are based <on tabulated overlap integrals data for Slaiter functions

Hybridization in 1,5,9-Tridehydro(l2)Annulene

The hybridization \u27in l,5,9-tridehydro(12)annulene has been calculated by the method of maximum overlap. The results show that the hybrids describing CC double bonds and CC triple bonds have more s-character than the hybrids of the same carbon atoms involved in single CC bonds. Consequently, CC double and CC triple bonds are strengthened not only by the additional n-type overlap, but also by an increase of cr-type overlap, which accompanies the rise of the s-content. By using available empirical correlations a comparison between calculated and experimental NMR data (spin-spin coupling constant J (C13- H) and chemical shift i;) is made

Hybridization in 1,5,9-Tridehydro(l2)Annulene

The hybridization \u27in l,5,9-tridehydro(12)annulene has been calculated by the method of maximum overlap. The results show that the hybrids describing CC double bonds and CC triple bonds have more s-character than the hybrids of the same carbon atoms involved in single CC bonds. Consequently, CC double and CC triple bonds are strengthened not only by the additional n-type overlap, but also by an increase of cr-type overlap, which accompanies the rise of the s-content. By using available empirical correlations a comparison between calculated and experimental NMR data (spin-spin coupling constant J (C13- H) and chemical shift i;) is made

Hybridization in Highly Strained Small Ring Hydrocarbons. II. Methylene Biscyclopropylidene and Dimethylene Biscyclopropylidenes

The hybridization in methylene biscyclopropylidene, 1,1\u27-dimethylene biscyclopropylidene, and 1,2-dimethylene biscyclopropylidene is considered by applying the method of maximum overlap. The results show variations in hybrids describing exocyclic double bonds: for ring carbons from sp~.ss to sp2-85, and for off-ring carbons from sp1-10 to sp1-41, depending on the immediate surounding. Bond overlaps for C=C consequently vary somewhat, the largest bond overlap being of the central C=C bond connecting two three membered rings. The results of the maximum overlap calculations are compared with available experimental data

Hybridization in 1,3,5-Cycloheptatriene and Some Related Molecules by the Method of Maximum Overlap

The hybridization in 1,3,5-cycloheptatriene, 1,3,5,7-cyclooctatetraene, 1,3-cyclohexadiene, and 1,4-cyclohexadiene has been calculated by the method of maximum overlap. The results show that if the molecules are assumed to be planar, hybrids describing the molecular skeleton deviate from the line joining the neighbouring carbon atoms and are directed towards the ins.ide of the ring. The deviation angles for the above molecules vary between 1.25° and 7 .5°. For non-planar structures the deviation angles are decreased or are equal to zero. The puckering of the molecular skeleton thus considerably reduces the strain

Hybridization in Highly Strained Small Ring Hydrocarbons. I. Tricyclo- and Tetracyclopropylidene

The hybridization in tricyclopropylidene and tetracyclopropylidene has been calculated by the method of maximum overlap. The results show that the hybrids describing the central ring have more s character than those in cyclopropane and cyclobutane: sp3·02 and sp3·31 as compared to sp3·86 and sp3·47 respectively. Consequently the central bonds in the two molecules investigated . are stabilized by the cyclopropyl substitutions. An interesting comparison between the maximum overlap hybridization and hybridization based on more elaborated calculations is made

Reduction of Hybrid Electron-repulsion Integrals to Overlap Integrals

It is shown how two-center electron-repulsion integrals of hybrid type can be reduced to linear combinations of overlap integrals. In certain cases the overlap integrals involve orbitals whose principal quantum numbers are less than their azimuthal quantum numbers. Explicit formulas are presented for the reduction of all hybrid integrals containing only orbitals of principal quantum numbers 1 or 2

Graph-theoretical Search for Benzenoid Polymers with Zero Energy Gap

Recently structural features which characterize the energy gap in polymeric conjugated hydrocarbons within the framework of the simple Hiickel MO theory have been specified using graphtheoretical methods. It has been shown that molecules of interest as potential units in polymers with zero energy gap have to satisfy certain structural conditions which relate the number of selected »excited« valence structures to that of the Kekule structures for the system. As in the previous work we consider units separated by essentially single bonds. Here we elaborate on the search for benzenoid systems which can satisfy the requirement. A necessary condition that a system may have zero energy gap is that the repeating benzenoid unit has a non-prime number (K) of Kekule valence structures. Then K could be factored: K1 • K2 ... Km. (Generally, there are more factorizations of K and our procedure needs to be carried out over all of them). Fragments Fi that have the number of Kekule structures given by K; are recognized, and we try to superimpose all of them over the skeleton of monomer unit. (Our procedure needs to be carried out for all possible fragmentations corresponding to a given factorization) . If it is possible to cover the monomer unit with all the fragments leaving at least two positions available for linking the units in polymer form, then the energy gap of such a polymer is zero. In a number of examples it is illustrated how the actual search is performed. A list of benzenoid systems of interest as units in a polymer with zero energy gap is given. The search is quite efficient and speedy

Hybridization in Several Polycyclic Alkanes by the Method of Maximum Overlap

The hybridization · in spiropentane, nortricyclene, cu bane, tricyclo( l.1.0.02,4)butane, bicyclo(l.1.l)pentane, and tetracyclo(3,3.l 2, 8.04,6) nonane has been determined by the method of maximum overlap. For the atomic functions Clementi Orbitals (ref. 1) have been assumed. A comparison between these results and those, obtained previously, by assuming Slater orbitals indicates the improvements that can be achieved by using more exact wave functions. The overlaps for CC bonds fall into the groups: (1) S 0.64. They are characteristic of highly strained three-membered rings, (6 ~ 20°), of four membered rings, (b ~ 10°), and of normal or very slightly bent bonds, (6 < 5°), respectively