  Diffusion Monte Carlo (DMC) calculations were performed for an accurate
description of the nature of the O$_2$ adsorption on a single layer graphene.
We investigated the stable orientation of O$_2$ at a specific adsorption site
as well as its equilibrium adsorption energy. At equilibrium adsorption
distances, an O$_2$ molecule was found to prefer a horizontal orientation,
where the O-O bond is parallel to the graphene surface, to the vertical
orientation. However, the vertical orientation is favored at the O$_2$-graphene
distances shorter than the equilibrium distance, which could be understood by
the steric repulsion between O and C atoms. Contrary to previous DFT
calculations, our DMC calculations show that the midpoint of a C-C bond (a
bridge site) is energetically preferred for the O$_2$ adsorption to a center of
a hexagonal ring (a hollow site). The lowest DMC adsorption energy was found at
an intermediate point between a hollow and a bridge site, where the O$_2$
adsorption energy was estimated to be -0.142(4) eV that was in very good
agreement with the recently-reported experimental value. Finally, we have found
that O$_2$ is very diffusive on the surface of graphene with the diffusion
barrier along a bridge-hollow-bridge path being as small as ~ 11 meV.
