Minimal model of inelastic tunneling of vibrating magnetic molecules on superconducting substrates

We present an efficient method of calculating the vibrational spectrum of a magnetic molecule adsorbed on a superconductor, directly related to the first derivative of the tunneling $IV$ curve. The work is motivated by a recent scanning-tunneling spectroscopy of lead phthalocyanine on superconducting Pb(100), showing a wealth of vibrational excitations, the number of which highly exceeds molecular vibrations typically encountered on normal metals. We design a minimal model, which represents the inelastic transitions by the spectral function of a frontier orbital of the molecule in isolation. The model allows for an exact solution; otherwise the full correlated superconducting problem would be hard to treat. The model parameters are supplied from an ab initio calculation, where the presence of the surface on the deformation of molecular geometry can be taken into account. The spectral function of the highest-occupied molecular orbital of the anionic ${\mathrm{PbPc}}^{1\ensuremath{-}}$ shows the best agreement with the experimental reference among other molecular charge states and orbitals. The method allows us to include multiple vibrational transitions straightforwardly.