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Graphene clusters consisting of $24-150$ carbon atoms and hydrogen termination at the zigzag
boundary edges have been studied,as well as clusters disordered by vacancy(s). Density Function
Theory and Gaussian03 software were used to calculate graphene relative stability,desorption energy,
band gap, density of states, surface shape, dipole momentum and electrical polarization of all clusters
by applying the hybrid exchange-correlation functional Beke-Lee-Yang-Parr. Further more,infrared
frequencies were calculated for two of them. Different basis sets,$6-31g^{**}$, $6-31g^{*}$ and $6-31g$,
depending on the sizes of clusters are considered to compromise the effect of this selection on the
calculated results. We found that relative stability and the gap decreases according to the size increase
of the graphene cluster.Mulliken charge variation increases with the size. For about 500 carbon atoms,
a zero HOMO-LUMO gap amount is predicted.Vacancy generally reduces the stability and having
vacancy affects the stability differently according to the location of vacancies. Surface geometry of each
cluster depends on the number of vacancies and their locations. The energy gap changes as with the
location of vacancies in each cluster. The dipole momentum is dependent on the location of vacancies
with respect to one another. The carbon-carbon length changes according to each covalence band
distance from the boundary and vacancies. Two basis sets,$6-31g^{*}$ and $6-31g^{**}$, predict equal amount
for energy, gap and surface structure,but charge distribution results are completely different.
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