  Scanning tunneling spectroscopy measures how a single electron with definite
energy propagates between a sample surface and the tip of a scanning tunneling
microscope. In the simplest description, the differential conductance measured
is interpreted as the local density of states of the sample at the tip
position. This picture, however, is insufficient in some cases, since
especially smaller molecules weakly coupled with the substrate tend to have
strong Coulomb interactions when an electron is inserted or removed at the
molecule. We present theoretical approaches to go from the non-interacting and
single-particle picture to the correlated many-body regime. The methodology is
used to understand recent experiments on finite armchair graphene nanoribbons
and phthalocyanines. We also theoretically discuss the strongly-correlated
model system of fractional quantum Hall droplets.
