We report the exact solution of the spectral problem for a graphene sheet framed by two armchair- and two zigzag-shaped boundaries. The solution is found for the pi electron Hamiltonian and gives, in particular, a closed analytic expression of edge-state energies in graphene. It is shown that the lower symmetry of graphene, in comparison with C6h of 2D graphite, has a profound effect on the graphene band structure. This and other results obtained have far-reaching implications for the understanding of graphene electronics. Some of them are briefly discussed.

An exact description of π electrons based on the tight-binding model of graphene as an alternant, plane macromolecule is presented. The model molecule can contain an arbitrary number of benzene rings and has armchair-and zigzag-shaped edges. This suggests an instructive alternative to the most commonly used approach, where the reference is made to the honeycomb lattice periodic in its A and B sublattices. Several advantages of the macromolecule model are demonstrated. The newly derived analytical relations detail our understanding of π electron nature in achiral graphene ribbons and carbon tubes and classify these structures as quantum wires.

The structural properties of self-assembled monolayers (SAMs) of oligo(ethylene glycol) (OEG)-terminated and amide-containing alkanethiols (HS(CH(2))(15)CONH(CH(2)CH(2)O)(6)H and related molecules with shorter alkyl or OEG portions) on gold are addressed. Optimized geometry of the molecular constituents, characteristic vibration frequencies, and transition dipole moments are obtained using density-functional theory methods with gradient corrections. These data are used to simulate IR reflection-absorption (RA) spectra associated with different OEG conformations. It is shown that the positions and relative intensities of all characteristic peaks in the fingerprint region are accurately reproduced by the model spectra within a narrow range of the tilt and rotation angles of the alkyl plane, which turns out to be nearly the same for the helical and all-trans OEG conformations. In contrast, the tilt of the OEG axis changes considerably under conformational transition from helical to all-trans OEG. By means of ab initio modeling, we also clarify other details of the molecular structure and orientation, including lateral hydrogen bonding, the latter of which is readily possessed by the SAMs in focus. These results are crucial for understanding phase and folding characteristics of OEG SAMs and other complex molecular assemblies. They are also expected to contribute to an improved understanding of the interaction with water, ions, and ultimately biological macromolecules.