Quantum properties of graphene and carbon nanostructures

In this thesis several physical and chemical properties of two dimensional graphene were investigated using the tight binding approximation and ab initio calculations. The dielectric functions are calculated with a quantum mechanical approach using the tight-binding approximation, this dielectric function describes the response of a graphene sheet, when an external electric field is applied. The Schrödinger equation was applied to graphene to describe the dynamical polarization in the structuring of the honeycomb lattice that depends on the Hamiltonian equation. The polarization function of graphene is calculated under electromagnetic radiation as a function of wave vector and frequency. At low wave vector the imaginary part increases with frequency approxiately quadratically. Furthermore, this procedure has been used to study the polarization function of asymmetric massless Dirac fermions at temperatures of 0, 77 and 300 K and a number of wave vectors.The calculate the wavefunction overlap of graphene asymmetric massless Dirac fermions was calculated by using the Hamiltonian equation. The results show that the imaginary part of the polarization function of asymmetric massless Dirac fermions when plotted as a function of wavevectors (𝑞𝑎), depends on the values of the anisotropy parameter 𝜆 and angle ∅(𝑞) at different temperatures. On the other hand the interaction between pyrazole, benzene and fluoro- and methylsubstituted benzenes as a molecules adsorbates on a single graphene sheets, modelled as an array of 6 x 6 fused aromatic rings of carbon and referred to as graphene (6,6) were also studied. The strength of the interaction were simulated using ab initio calculations with dispersion corrected density functional theory (DFT). The 𝜔B97XD/ 6-31G(d,p) functional/basis combination was employed to probe the interaction as a function of the distance between the graphene sheet and the aromatic species, along with the faster semi-empirical PM6 method to provide a comparison to a lower level, non-dispersion corrected method. Furthermore, this procedure has been used to study the interaction between benzene and fluoro- and methyl- substituted benzenes with graphene (5,5) - a smaller 5 x 5 array of fused aromatic rings. Given that isomeric forms may exist for some of the substituted benzenes, the effect of isomeric forms on the binding of the adsorbate to the graphene has also been studied in this work. The energetics of the interaction between graphene and the aromatic molecule were restrained in a parallel geometry and are largely van der Waals’ in nature. At separation distances greater than that corresponding to the minimum energy, which was found to be dependent upon the precise nature of the interacting aromatic species, the interaction was found to become increasingly weak with increasing separation between the graphene sheet and aromatic species. By contrast, closer approaches were found to result in pronounced repulsive forces. Significant differences in the energies of interaction were observed when changing the group substituents from hydrogen to fluorine or methyl groups. The degree of charge transfer between the graphene and adsorbate was also investigated using density functional theory (DFT). The 𝜔B97X-D and B3LYP/6- 31+G(d,p) and 6-31++G(d,p) functional/basis. Using Hirshfeld charge analysis, Atoms in Molecules (Bader) and Mulliken population analysis, the charge transfer between adsorbed molecules and a graphene surface, showed a significant dependence on the charge analysis method. In general, the degree of charge transfer was very small in this class of complexes, corresponding to only a small fraction of an electron.