of the harmonic oscillator only provides an incoherent background with a small gap at zero frequency remaining. In this case the elastic contributions stemming from the side peaks in $\rho_{d_\sigma,d^\dagger_\sigma}(\w)$ could be mistakenly attributed to inelastic contributions stemming from a fictitious phonon at frequency $\w_0' = |U_{\rm eff}|$.

Summary and conclusion {#sec:conclusion}
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We extended the tunnel theory for STS that bridges between weak and strong electron phonon coupling in the system as well as includes the strong coupling limit of electronic degrees of freedom. It relies on an exact solution of all relevant system spectral functions of the system in the absence of the STM tip and requires only that the coupling to the tip remains weak so that the second order expansion in the tunneling matrix element is sufficient. Importantly, a Wick’s theorem is not required.

One of the key incidence of our approach is the systematic derivation of the tunnel current operator from the charge conservation in the total system comprising the system S of interest and the STM tip. The analytic form of the current operator is determined by the tunneling Hamiltonian $H_T$ connecting the two parts of the total system. The strength of the approach is the treatment of all tunneling processes, elastic and inelastic contributions, on equal footing. In particular, our approach includes linear as well as quadratic contributions to the inelastic tunneling current. Neglecting the linear term is only applicable in the limit of vanishing electron-phonon coupling in the system. It becomes relevant in situations when the phonon mode not only enters $H_T$ but couples to the electronic degrees of freedom in the system S.

We presented experimental STS for two different location of NTCDA molecules on Ag(111). Combining the LDA+MBPT with an NRG approach clarifies that the so-called bright molecule corresponds to the top molecule and the dark molecule corresponds to the bridge molecule in the ab-initio calc