In search of Coulson’s lost theorem

In Hückel theory, the bond number is the sum of the orders of the π bonds incident on a given carbon center. From the work of Coulson and his school, it has been believed for over 70 years that the bond number has a maximum of 3⎯⎯√ and that this bound is realized by exactly one conjugated framework, that of the trimethylenemethane radical. Search of published literature and archived correspondence failed to find any formal proof of these two statements. Here, we provide a new formula for bond number that leads to an easily checked proof of both. The bond number of graphene is 1.574 597… (90.9% of the mathematical limit), and this value appears to act as a separator for the classes of metallic and semiconducting single-walled nanotubes, as defined within Hückel theory

Topological Anisotropy of Stone-Wales Waves in Graphenic Fragments

Stone-Wales operators interchange four adjacent hexagons with two pentagon-heptagon 5|7 pairs that, graphically, may be iteratively propagated in the graphene layer, originating a new interesting structural defect called here Stone-Wales wave. By minimization, the Wiener index topological invariant evidences a marked anisotropy of the Stone-Wales defects that, topologically, are in fact preferably generated and propagated along the diagonal of the graphenic fragments, including carbon nanotubes and graphene nanoribbons. This peculiar edge-effect is shown in this paper having a predominant topological origin, leaving to future experimental investigations the task of verifying the occurrence in nature of wave-like defects similar to the ones proposed here. Graph-theoretical tools used in this paper for the generation and the propagation of the Stone-Wales defects waves are applicable to investigate isomeric modifications of chemical structures with various dimensionality like fullerenes, nanotubes, graphenic layers, schwarzites, zeolites

OH-functionalized open-ended armchair single-wall carbon nanotubes (SWCNT) studied by density functional theory

The structures of ideal armchair (5,5) single-wall carbon nanotubes (SWCNTs) of different lengths (3.7, 8.8, and 16.0 Å for C40H20, C80H20, and C140H20) and with 1–10 hydroxyl groups at the end of the nanotube were fully optimized at the B3LYP/3-21G level, and in some cases at the B3LYP/6-31G* level, and the energy associated with the attachment of the OH substituent was determined. The OH-group attachment energy was compared with the OH functionalization of phenanthrene and picene models and with previous results for zigzag (9.0) SWCNT systems. In comparison to zigzag SWCNTs, the armchair form is more (by about 5 to 10 kcal mol−1) reactive toward hydroxylation

On the Zwitterionic Nature of Gas-Phase Peptides and Protein Ions

Determining the total number of charged residues corresponding to a given value of net charge for peptides and proteins in gas phase is crucial for the interpretation of mass-spectrometry data, yet it is far from being understood. Here we show that a novel computational protocol based on force field and massive density functional calculations is able to reproduce the experimental facets of well investigated systems, such as angiotensin II, bradykinin, and tryptophan-cage. The protocol takes into account all of the possible protomers compatible with a given charge state. Our calculations predict that the low charge states are zwitterions, because the stabilization due to intramolecular hydrogen bonding and salt-bridges can compensate for the thermodynamic penalty deriving from deprotonation of acid residues. In contrast, high charge states may or may not be zwitterions because internal solvation might not compensate for the energy cost of charge separation

DFT study on the cycloaddition reactions of [c]-annelated carbo- and heterocyclic five-membered dienes with ethylene

B3LYP/6-31G* calculations were done on a series of [c]-annelated heterocyclic five-membered dienes and their cycloaddition transition state structures and products. Cyclobutano[c]-, cyclobuteno[c]- and benzo[c]-fused five-membered rings represent non-aromatic, antiaromatic and aromatic ring fused dienes, respectively, and their reactivity was studied with ethylene as dienophile. In the cases of cyclobutano- and cyclobuteno-fused dienes, the fused portion remains a mere spectator and no significant geometric variations are seen along the reaction coordinate. In contrast, in the benzo-fused systems, the fused benzene ring witnesses significant changes in the bond lengths along the reaction coordinate highlighting the active participation of the π-framework in the reaction. Thus, this may be classified as an [8 + 2] cycloaddition. The benzo[c]-fused rings exhibit the lowest activation energies followed by cyclobutano and cyclobuteno analogues, and the exothermicities decrease in the same order. A linear relationship between the reaction exothermicity and activation energy is obtained for the systems under consideration. The percentage bond length alternation on going from reactant to transition state, the Fukui function indices and frontier orbital analysis, extent of charge transfer from diene to dienophile at the transition state, and deformation energies were estimated to explain the reactivity of the dienes with ethylene