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The setup is illustrated in Fig. 4 where a small bias voltage V$_{B}$ ∼ Γ /e has been applied and a gate voltage V$_{G}$ has been tuned such that the position of the reso- nance is located ℏ ω above the bias window. It is crucial that the electronic resonance has a width much smaller than the quantum of vibration (Γ ≪ ℏ ω ), since otherwise there will be a small but constant transmission probabil- ity when the molecule is in the vibrational ground state. 

As an example we have performed a density func- tional theory (DFT) study of a benzene molecule sand- wiched between two Au(111) contacts. The calculations were performed with the code gpaw 16,17 which is a real- space DFT code using the projector augmented wave method $^{18}$. The contact were simulated by a three layer Au(111) slab where the the top layer has been relaxed. We used a supercell with 12 Au atoms in each slab layer which were sampled by a 4 × 4 grid of K-points and 12 . 2 A of vacuum. Benzene were added with its plane parallel to the surface and the adsorption energy was calculated as a function of distance to the slab. This is shown in Fig. 5 for PBE $^{19}$, revPBE20 and vdW 21 functionals. The PBE and revPBE functional shows weak and no bonding respectively whereas the vdW functional shows a 0 . 3 eV minimum at 3 . 6 A . The weak van der Waals bonding indicates that benzene is physisorbed on Au(111) which means that the molecular orbitals are weakly hybridized with metallic states as required by vibrationally medi- ated transmission of single electrons. This can also be seen explicitly from the Kohn-Sham projected density of states from which we estimate the width of HOMO and LUMO resonances to be Γ$_{L}$ ∼ 0 . 01 eV . It should be 

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noted that we assumed a LUMO state in Eq. (1) and Fig. 2, but the analysis is equally valid for transmis- sion of hole states mediated by the HOMO and in the following we will consider both types of resonance. Ben- zene has two degenerate HOMOs and two degenerate LU- MOs which have the potential to mediate transmission of electrons through the molecule. One of the LUMOs is shown in Fig. 6 and it is expected that a transient occupation of the orbital may induce internal forces in the molecule and thus couple to the vibrational modes of the molecule. We have performed a DFT-based normal mode analysis of the benzene molecule which has 36 vi- brational modes. There are 6 degenerate highly energetic modes with ℏ ω$_{i}$ ∼ 0 . 39 eV and the rest of the modes are evenly distributed in the interval ℏ ω$_{i}$ ∼ 0 − 0 . 20 eV . The high energy modes involve the hydrogen atoms oscillat- ing in the plane of the molecule along the individual H-C bonds as shown in Fig. 6. The HOMO and LUMO states are expected to couple to several of the molecular vibra- tional states but we will focus on the high energy modes which have highly separated vibrational sidebands in the weak coupling limit. The coupling constants λ$_{i}$ can be related to the excited state potential energy surface V$_{a}$ associated with the LUMO being occupied or the HOMO being emptied $^{22}$: 

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where u$_{i}$ is the coordinate of the i ’th normal mode, u 0 i its equilibrium position in the electronic ground state, m$_{i}$ its effective mass and V$_{a}$ the excited state potential energy surface associated with the resonance $^{22,23}$. To obtain V$_{a}$ we have used the method of linear expansion Δ self-consistent field DFT 23 which allows us to calcu- late the excited state energies while moving the atoms along the high energy mode where the 6 hydrogen atoms move in phase (see Fig. 6). The results for this mode 

