  A microscopic theory of the transport in a scanning tunnelling microscope
(STM) set-up is introduced for \pi-conjugated molecules on insulating films,
based on the density matrix formalism. A key role is played in the theory by
the energy dependent tunnelling rates which account for the coupling of the
molecule to the tip and to the substrate. In particular, we analyze how the
geometrical differences between the localized tip and extended substrate are
encoded in the tunnelling rate and influence the transport characteristics.
Finally, using benzene as an example of a planar, rotationally symmetric
molecule, we calculate the STM current voltage characteristics and current maps
and analyze them in terms of few relevant angular momentum channels.
