  In this work we study STM signal in the presence of Coulomb assisted hopping.
We perform an extended two site Hubbard model analysis between the atom on the
tip and the atom in the sample nearest to each other. We show that in the
presence of Coulomb assisted hopping the STM signal depends on several spectral
functions thereby complicating its interpretation. Furthermore in the broadband
tip limit there are now three different competing rates for the total current
(instead of one for the usual two site Hubbard model analysis used in the
literature so far). We find an exact (within the Fermi golden rule - that is in
the limit of weak coupling between tip and sample) expression for the current
as a function of the bias voltage. As an example we apply our calculations to
the case of free fermions with a uniform density of states. Even in this simple
case there are non-trivial corrections - where the $dI/d\mathcal{V}$ (the rate
of change of the current with respect to bias voltage) is not uniform as a two
site (non-extended) Hubbard model analysis would predict. We also show that for
realistic conditions the corrections predicted here are order one.
