The rich and unique properties of the stage-n graphite alkali-metal-intercalation compounds are fully investigated by first-principles calculations.

We use first-principles calculations within the density functional theory (DFT) to explore the electronic properties of stage-1 Li- and Li+-graphite-intercalation compounds (GIC) for different concentrations of LiCx/Li+Cx, with x = 6, 12, 18, 24, 32 and 36. The essential properties, e.g., geometric structures, band structures and spatial charge distributions are determined by the hybridization of the orbitals, the main focus of our work. The band structures/density of states/spatial charge distributions display that Li-GIC shows a blue shift of Fermi energy just like metals, but Li+-GIC still remains as in the original graphite or exhibits so-called semi-metallic properties, possessing the same densities of free electrons and holes. According to these properties, we find that there exist weak but significant van der Waals interactions between interlayers of graphite, and 2s-2pz hybridization between Li and C. There scarcely exist strong interactions between Li+-C. The dominant interaction between the Li and C is 2s-2pz orbital-orbital coupling; the orbital-orbital coupling is not significant in the Li+ and C cases, but dipole-diploe coupling is.

How to form carbon nanoscrolls with non-uniform curvatures is worthy of a detailed investigation. The first-principles method is suitable for studying the combined effects due to the finite-size confinement, the edge-dependent interactions, the interlayer atomic interactions, the mechanical strains, and the magnetic configurations. The complex mechanisms can induce unusual essential properties, e.g., the optimal structures, magnetism, band gaps and energy dispersions. To reach a stable spiral profile, the requirements on the critical nanoribbon width and overlapping length will be thoroughly explored by evaluating the width-dependent scrolling energies. A comparison of formation energy between armchair and zigzag nanoscrolls is useful in understanding the experimental characterizations. The spin-up and spin-down distributions near the zigzag edges are examined for their magnetic environments. This accounts for the conservation or destruction of spin degeneracy. The various curved surfaces on a relaxed nanoscroll will create complicated multi-orbital hybridizations so that the low-lying energy dispersions and energy gaps are expected to be very sensitive to ribbon width, especially for those of armchair systems. Finally, the planar, curved, folded, and scrolled graphene nanoribbons are compared with one another to illustrate the geometry-induced diversity.

We study the number of dimer-monomers M d (n) on the Tower of Hanoi graphs T H d (n) at stage n with dimension d equal to 3 and 4. The entropy per site is defined aswhere v is the number of vertices on T H d (n). We obtain the lower and upper bounds of the entropy per site, and the convergence of these bounds approaches to zero rapidly when the calculated stage increases. The numerical value of z T H d is evaluated to more than a hundred digits correct. Using the results with d less than or equal to 4, we predict the general form of the lower and upper bounds for z T H d with arbitrary d.The dimer-monomer model is an interesting but elusive model in statistical mechanics [1][2][3]. In this model, a dimer is realized by a diatomic molecule with two neighboring sites attaching to a surface or lattice. For the sites that are not occupied by any dimers, they could be regarded as covered by monomers. Let us define N DM (G) to be the number of dimer-monomers on a graph G.The computation of the general dimer-monomer model remains to be a difficult problem [4], in contrast to the closed-packed dimer problem on planar lattices that had been discussed and solved more than fifty years ago [5][6][7]. Recent computation of close-packed dimers, dimers with a single monomer, and general dimer-monomer models on regular lattices are given in Refs. [8][9][10][11][12][13][14][15][16][17][18]. It is also interesting to discuss the dimer-monomer problem on fractals with scaling invariance but not translational invariance. The fractals with noninteger Hausdorff dimension can be constructed from certain basic shape [19,20]. A famous fractal is the Tower of Hanoi graph, and it has been discussed in different contexts [21][22][23].The dimer-monomer problem on the Tower of Hanoi graph with dimension d = 2 was discussed in [24]. In this article, we shall first recall some basic definitions in section II. In section III, we present the recursion relations for the number of dimer-monomers on T H d (n) with dimension d = 3, then enumerate the entropy per site using lower and upper bounds in details. The calculation for T H d (n) with dimension d = 4 will be given in section IV. In the last section, we shall predict the general form of the lower and upper bounds of the entropy per site for dimer-monomers on the Tower of Hanoi graph with arbitrary dimension. II. PRELIMINARIESIn this section, let us review some basic terminology. A graph G = (V, E) that is connected and has no loops is defined by the vertex (site) set V and edge (bond) set E [25,26].Denote v(G) = |V | as the number of vertices in G and e(G) = |E| as the number of edges.Two vertices a and b are neighboring if the edge ab is included in E. A matching of a graph G is an independent edge subset where the edges have no common vertices. The number of matching in G is denoted as N DM (G), which corresponds to the number of dimer-monomers in statistical mechanics. Although monomer and dimer weights can be associated to each

We present the number of dimer coverings Nd(n) on the Tower of Hanoi graph THd(n) at n stage with dimension 2 [Formula: see text][Formula: see text]d[Formula: see text][Formula: see text] 5. When the number of vertices v(n) is even, Nd(n) gives the number of close-packed dimers; when the number of vertices is odd, it is impossible to have a close-packed configurations and one of the outmost vertices is allowed to be unoccupied. We define the entropy of absorption of diatomic molecules per vertex as S[Formula: see text][Formula: see text]=[Formula: see text][Formula: see text] Nd(n)/v(n), that can be shown exactly for TH2, while its lower and upper bounds can be derived in terms of the results at a certain n for THd(n) with 3 [Formula: see text][Formula: see text]d[Formula: see text][Formula: see text] 5. We find that the difference between the lower and upper bounds converges rapidly to zero as n increases, such that the value of S[Formula: see text] with d[Formula: see text]=[Formula: see text]3 and 5 can be calculated with at least 100 correct digits.