  We have carried out the systematic study of quantum capacitance (C$_Q$) in
functionalized graphene. The functionlization of graphene has been done by
doping with different aliphatic and aromatic molecules and their radicals.
Using density functional theory calculations, we first analyze the electronic
band structure of the functionalized graphene and subsequently obtained the
quantum capacitance in each system. We observed that the quantum capacitance
can be enhanced by doping aliphatic and aromatic molecules and their radicals
on graphene sheet, especially radical functionalized graphene shows significant
enhancement in C$_Q$. Our theoretical investigation reveals that aromatic and
aliphatic radicals generates localized density of states near the Fermi level,
due to a charge localization. As a result, a very high quantum capacitance
(above 200 $\mu F/cm^2$) compared to pristine graphene has been estimated.
Effects of atomic dislocation and the vacancy defect on graphene during
functionalization has also been incorporated in our investigation. Obtain
results predict that the formation of molecular radicals while
functionalization of graphene could be an efficient way to synthesize highly
efficient graphene based supercapacitor electrode materials which in turn may
significantly improve the performance of supercapacitors.
