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The thermoelectric transport through a ring threaded by an Aharonov-Bohmflux, with a molecular bridge on one of its arms, is analyzed. The transportelectrons also interact with the vibrational excitations of that molecule. Thisnano-system is connected to three terminals: two are electronic reservoirs,which supply the transport electrons, and the third is the phonon bath whichthermalizes the molecular vibrations. Expressions for the transportcoefficients, relating all charge and heat currents to the temperature andchemical potential differences between the terminals, are derived to secondorder in the electron-vibration coupling. At linear response, all thesecoefficients obey the full Onsager-Casimir relations. When the phonon bath isheld at a temperature different from those of the electronic reservoirs, a heatcurrent exchanged between the molecular vibrations and the transport electronscan be converted into electric and/or heat electronic currents. The relatedtransport coefficients, which exist only due to the electron-vibrationcoupling, change sign under the interchange between the electronic terminalsand the sign change of the magnetic flux. It is also demonstrated that theAharonov-Bohm flux can enhance this type of conversion.

Three-terminal thermoelectric transport through a molecule placed on an Aharonov-Bohm ring