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We present an extension of the tunneling theory for scanning tunnelingmicrocopy (STM) to include different types of vibrational-electronic couplingsresponsible for inelastic contributions to the tunnel current in thestrong-coupling limit. It allows for a better understanding of more complexscanning tunneling spectra of molecules on a metallic substrate in separatingelastic and inelastic contributions. The starting point is the exact solutionof the spectral functions for the electronic active local orbitals in theabsence of the STM tip. This includes electron-phonon coupling in the coupledsystem comprising the molecule and the substrate to arbitrary order includingthe anti-adiabatic strong coupling regime as well as the Kondo effect on a freeelectron spin of the molecule. The tunneling current is derived in second orderof the tunneling matrix element which is expanded in powers of the relevantvibrational displacements. We use the results of an ab-initio calculation forthe single-particle electronic properties as an adapted material-specific inputfor a numerical renormalization group approach for accurately determining theelectronic properties of a NTCDA molecule on Ag(111) as a challenging samplesystem for our theory. Our analysis shows that the mismatch between theab-initio many-body calculation of the tunnel current in the absence of anyelectron-phonon coupling to the experiment scanning tunneling spectra can beresolved by including two mechanisms: (i) a strong unconventional Holstein termon the local substrate orbital leads to reduction of the Kondo temperature and(ii) a different electron-vibrational coupling to the tunneling matrix elementis responsible for inelastic steps in the $dI/dV$ curve at finite frequencies.

Inelastic electron tunneling spectroscopy for probing strongly correlated many-body systems by scanning tunneling microscopy