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The total ground state energy and electronic band structure of Graphite and Diamond were calculated in this work using FHI-aims (Fritz Haber Institute ab initio molecular simulations) Density Functional Theory (DFT) code. The density functionals used are the generalised gradient functional PBE, and PBE+vdW approach as defined by Tkatchenko and Scheffler. The results obtained from the computations of the ground state energies of diamond and graphite were -2056.898408114 eV and -2061.703700984 eV respectively. Similarly, the results obtained from the calculations of the electronic band gaps of graphite and diamond were 0.0 eV and 5.56369215 eV, respectively. These are in good agreement when compared to the experimental values of 0eV and 5.48eV. These band gaps are within reasonable percentage errors of 0.0% and 1.46% respectively. This shows that DFT has overestimated the band gap of diamond by 0.08 eV in our computations.