**5. Conclusions**

In this paper, we have theoretically analyzed the role of electron–vibration and electron–electron interactions on the thermoelectric properties of molecular junctions focusing on devices based on fullerene. We have used a self-consistent adiabatic approach which allows a non-perturbative treatment of the electron coupling to low frequency vibrational modes, such as those of the molecular center of mass between metallic electrodes. This approach incorporates Coulomb blockade effects due to strong electron–electron interaction between molecular degrees of freedom. We have analyzed a one-level model which takes into account the LUMO level of fullerene and its alignment to the chemical potential. We have stressed that an accurate description of the transport properties is obtained in the intermediate regime for the electron–vibration coupling and in the strong coupling regime for the electron–electron interaction. Moreover, we have demonstrated that only the combined effect of electron–vibration and electron–electron interactions is able to predict the correct behavior of both the charge conductance and the Seebeck coefficient. The theoretical calculations presented in this paper show a very good agreemen<sup>t</sup> with available experimental data of both charge conductance and Seebeck coefficient.

In this paper, we have used a one-level transport model as a starting point to address the role of many-body interactions between molecular degrees of freedom. This model is frequently used in all cases where the energy levels can be tuned around the chemical potential and additional spectral features are absent [4,17]. This is the case of the experiments in [17] for the fullerene junctions analyzed in this paper. The one-level model is expected to be valid for energies close to the Fermi level and low temperatures. Actually, a more realistic description of the molecule and its coupling with metallic leads is needed if more complex transport phenomena take place, in particular interference effects [47,48] recently investigated in molecular junctions. Moreover, inclusion of quantum corrections to oscillator dynamics can be important to explore the effects of additional vibrational modes and further electron–vibration regimes [49] (from adiabatic to anti-adiabatic ones) and their relation with strong electron–electron interactions [50,51].

**Author Contributions:** Writing—original draft preparation, C.A.P., Supervision, V. C.

**Funding:** C.A.P. was funded by project Fondo per il finanziamento delle attivita base di ricerca anno 2018. **Conflicts of Interest:** The authors declare no conflict of interest.
