We present a reformulation of the Hairy Probe method for introducing electronic open boundaries
that is appropriate for steady state calculations involving non-orthogonal atomic basis sets. As a
check on the correctness of the method we investigate a perfect atomic wire of Cu atoms, and a
perfect non-orthogonal chain of H atoms. For both atom chains we find that the conductance has
a value of exactly one quantum unit, and that this is rather insensitive to the strength of coupling
of the probes to the system, provided values of the coupling are of the same order as the mean
inter-level spacing of the system without probes. For the Cu atom chain we find in addition that
away from the regions with probes attached, the potential in the wire is uniform, while within
them it follows a predicted exponential variation with position. We then apply the method to an
initial investigation of the suitability of graphene as a contact material for molecular electronics.
We perform calculations on a carbon nanoribbon to determine the correct coupling strength of the
probes to the graphene, and obtain a conductance of about two quantum units corresponding to
two bands crossing the Fermi surface. We then compute the current through a benzene molecule
attached to two graphene contacts and find only a very weak current because of the disruption of
the π-conjugation by the covalent bond between the benzene and the graphene. In all cases we find
that very strong or weak probe couplings suppress the current.