To describe the interaction of molecular vibrations with electrons at a quantum dot contacted to metallic leads, we extend an analytical approach that we previously developed for the many-polaron problem. Our scheme is based on an incomplete variational Lang-Firsov transformation, combined with a perturbative calculation of the electron-phonon self-energy in the framework of generalised Matsubara functions. This allows us to describe the system at weak to strong coupling and intermediate to large phonon frequencies. We present results for the quantum dot spectral function and for the kinetic coefficient that characterises the electron transport through the dot. With these results we critically examine the strengths and limitations of our approach, and discuss the properties of the molecular quantum dot in the context of polaron physics. We place particular emphasis on the importance of corrections to the concept of an antiadiabatic dot polaron suggested by the complete Lang-Firsov transformation.
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