Hückel spectra of Möbius π systems

Electron-electron interactions and topology in the electronic properties of gated graphene nanoribbon rings in Möbius and cylindrical configurations

“…2(b)], the number of atoms in a chain M must be even in order to keep the hexagonal structure intact. The analytical solutions to the cyclic and Möbius structures were previously calculated, 49,50 and are given by…”

“…10 Interestingly, it has been suggested that nontrivial properties of graphene nanoribbons can be generated directly by engineering a nontrivial Möbius geometry of the nanoribbon without the need for the spin-orbit coupling. [40][41][42][43][44][45][46][47][48][49][50][51] For example, it was shown by Guo et al 46 that one electron states of a class of Möbius graphene ribbons with zigzag edges can be understood by introducing a non-Abelian gauge field 46 as in topological insulators. [6][7][8][9][10][11][12] However, these authors used the s orbitals instead of p z orbitals as the basis needed to describe graphene nanoribbons and limited their work to an even number of atomic chains.…”

Geometric localisation in Möbius π systems

“…Eigenvalue spectra of Möbius r and p systems are thus in general different. In fact, as the Mö bius band is doubly covered by the cylinder (in the sense that identification of antipodal points on a cylinder produces a set of points with the topology of the band [2]), the eigenvalue spectra of r and p systems for a fixed Mö bius carbon skeleton form complementary halves of the spectrum of a centrosymmetric graph with twice as many vertices, a graph that can be embedded on the cylinder [23]. Gerade eigenvectors of the cylindrical graph give r, ungerade p, solutions for the smaller Mö bius graph.…”

“…Gerade eigenvectors of the cylindrical graph give r, ungerade p, solutions for the smaller Mö bius graph. Rules for stability of 4n + 2 and 4n cycles reverse between Hü ckel and Mö bius systems, and the mnemonic constructions such as the Frost-Musulin [24] diagram for the eigenvalue spectra need modification, but the double-cover property allows both Hückel and Mö bius cycles to be included in a single diagram for the cycle with double the number of vertices [23].…”

Spectra and structural polynomials of graphs of relevance to the theory of molecular conduction

“…We therefore include certain graphs with edge weights having flipped signs, and/or with a pair of crossed backbone edges. These flipped and crossed graphs are sometimes called Möbius graphs [6]. The cases that we consider here have closed-form spectra and structural polynomials that can be derived using the methodology used for unflipped, uncrossed graphs.…”