Transition metal dichalcogenides (TMDCs) have emerged as a new two-dimensional material's field since the monolayer and few-layer limits show different properties when compared to each other and to their respective bulk materials. For example, in some cases when the bulk material is exfoliated down to a monolayer, an indirect-to-direct band gap in the visible range is observed. The number of layers N (N even or odd) drives changes in space-group symmetry that are reflected in the optical properties. The understanding of the space-group symmetry as a function of the number of layers is therefore important for the correct interpretation of the experimental data. Here we present a thorough group theory study of the symmetry aspects relevant to optical and spectroscopic analysis, for the most common polytypes of TMDCs, i.e., 2H a, 2H c and 1T , as a function of the number of layers. Real space symmetries, the group of the wave vectors, the relevance of inversion symmetry, irreducible representations of the vibrational modes, optical activity, and Raman tensors are discussed.

We investigate the energetic, structural, electronic and thermodynamics properties of hydrogen fluoride cluster, (HF)n, in the range n=2–8, by ab initio methods and density functional theory (DFT). The ab initio methods chosen were Hartree–Fock (RHF) and second-order Mo/ller–Plesset perturbation theory (MP2). The DFT calculations were based on Becke’s hybrid functional and the Lee–Yang–Parr correlation functional (B3LYP). We found that symmetric cyclic clusters are the most stable structure, and that large cooperative effects, particularly from trimer to tetramer are present, in binding energy, and hydrogen bond distance. An analysis of the topology of the electron density reveals a linear correlation between the binding energy per hydrogen bond and the density at the hydrogen bond critical point and the Cioslowski covalent bond order. Based on these correlations, hydrogen bond cooperativity is associated with the electronic delocalization between monomers units. Analysis of the thermodynamics properties shows that the enthalpy changes are determined by the electronic cooperative effects, while the entropic statistical factors are fundamental in the relative stability of these clusters. Finally, for the trimer and tetramer, nonstable linear zigzag chains where found in a detailed analysis of the potential energy surfaces.

Group theory analysis for two-dimensional elemental systems related to phosphorene is presented, including (i) graphene, silicene, germanene and stanene, (ii) their dependence on the number of layers and (iii) their two possible stacking arrangements. Departing from the most symmetric D 1 6h graphene space group, the structures are found to have a group-subgroup relation, and analysis of the irreducible representations of their lattice vibrations makes it possible to distinguish between the different allotropes. The analysis can be used to study the effect of strain, to understand structural phase transitions, to characterize the number of layers, crystallographic orientation and nonlinear phenomena.

ABSTRACT:This article studies the cooperativity present in hydrogen fluoride clusters, (FH) n , by means of a many-body decomposition of the binding energy. With the aim of quantifying how the results depend on the calculation level, the partition was performed from dimer to hexamer at the RHF, MP2, and density functional (B3LYP) levels, and for the heptamer and octamer at the RHF and B3LYP levels, using a 6-31ϩϩG(d, p) basis set in all cases. We obtain that, for a proper representation of the cooperative effects in hydrogen fluoride, at least the inclusion of the three-body terms is fundamental. The contributions are found to be underestimated at the RHF level and overestimated at the B3LYP level, with respect to the MP2 results.