3 h R L B eV G V FIG. 5: Adsorption energy of benzene on Au(111) as a func- tion of distance to the surface, calculated with three different functionals. The molecular states are weakly hybridized with the metallic states and only the functional including the van der Waals interaction gives the correct adsorption well. FIG. 4: The principle of single electron transmission. A small bias is applied such that the bias window just covers the res- onance: eV B ∼ Γ and a gate voltage is tuned such that the resonance is located at: ε 0 ∼ µ L − eV B / 2 + ℏ ω . In the vibra- tional ground state the transmission function T 00 (solid line) is zero in the bias window. Exciting the first vibrational state changes the transmission function which is dominated by the inelastic part T 10 (dashed line) in the bias window. A vibra- tional excitation of the molecule will thus result in a single electron being transmitted. 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 $^{22}$: 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 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 λ$_{i}$ = l$_{i}$ √$_{2}$ ∂V$_{a}$ ∂u$_{i}$ ∣ u$_{i}$ = u 0 i , l$_{i}$ = √ ℏ m$_{i}$ω$_{i}$ , (3)