4 Appendix A T$_{n}$ ( ε$_{i}$, ε$_{f}$ ) =Γ$_{L}$Γ$_{R}$ ∫ dτdtds 2 π ℏ 3 e i [( ε$_{i}$ − ε$_{f}$ ) τ + ε$_{f}$t − ε$_{i}$s ] / ℏ × G$_{n}$ ( τ, s, t ) , (A1) G$_{n}$ ( τ, s, t ) = θ ( s ) θ ( t ) 〈 n | c$_{a}$ ( τ − s ) c † $_{a}$( τ ) c$_{a}$ ( t ) c † a | n 〉 (A2) G$_{n}$ ( τ, s, t ) = G 0 $_{R}$( t ) G 0 ∗ R ( t ) e igω 0 ( t − s ) × e − $^{gf$_{τ,s,t}$}$L$_{n}$ [ g ( f$_{τ,s,t}$ + f ∗ $_{τ,s,t}$)] , (A3) G 0 $_{R}$( t ) = − iθ ( t ) e − i ( ε 0 − i Γ / 2) t/ $^{ℏ}$, f$_{τ,s,t}$ = 2 − e − iω 0 t − e iω 0 s + e − iω 0 $^{τ}$(1 − e iω 0 $^{t}$)(1 − e iω 0 $^{s}$) . FIG. 6: Left: The lowest unoccupied orbital (LUMO) of ben- zene. The red and blue surfaces are positive and negative isosurfaces of the wavefunction. Right: The vibrational mode of highest energy which can be excited by a transient occu- pation of the LUMO. mum requirement for the principle to work. However, there may be others reasons why this system is not well suited for experiments of this kind and it would be very interesting to investigate the principle in systems where transmission through single molecules with significant vi- brational coupling has been observed $^{1–5}$. We are grateful to K. S. Thygesen for advice and com- ments on the manuscript. This work was supported by the Danish Center for Scientific Computing. The Cen- ter for Individual Nanoparticle Functionality (CINF) is sponsored by the Danish National Research Foundation. In this appendix we will show the details of the cal- culations leading to Fig. 2 for a single vibrational mode of frequency ω$_{0}$ and coupling λ$_{0}$ . Within scattering the- ory the transmission matrix T$_{n}$ ( ε$_{i}$, ε$_{f}$ ) for a vibrationally excited state n can be expressed as 6 where is the two particle Green function of the vibrational state n . The Green function can be evaluated exactly in the wide band limit and the result is 15 are λ HOMO = 27 meV and λ LUMO = 9 meV which should be compared with the quantum of oscillation ℏ ω$_{i}$ ∼ 0 . 39 eV . The coupling is thus rather weak and we obtain maximum probabilities of inelastic transmission of 5 × 10 − 3 and 5 × 10 − 4 respectively (with Γ = 0 . 01 eV ) at the lower vibrational sideband. However, assuming the vibrational lifetime to be on the order of nanosec- onds, the probabilities are most likely large enough that an elastic transmission event will occur. Furthermore, a small coupling constant means that the ratio T$_{11}$/T$_{10}$ becomes very small and vibrational excitation will thus nearly always result in only a single electron being trans- mitted. where L$_{n}$ is the n ’th Laguerre polynomial, g = λ 2 $_{0}$/ ( ℏ ω$_{0}$ ) $^{2}$, and In summary, we have presented a method, which al- lows one to control the transmission of single electrons in weakly coupled molecule-metal junctions. The trans- mission is mediated by exciting a vibrational mode of the molecule while a gate voltage is tuned such that the reso- nant state is kept a quantum of vibrational energy above the bias window. It is assumed that such an excitation can be obtained with an external perturbation, for exam- ple a short laser pulse. The requirement of weak metallic coupling (Γ ≪ ℏ ω ) is essential since it excludes elastic transmission in the vibrational ground state. For small vibrational coupling ( λ ≪ ℏ ω ) the junction will then be highly reliable and always give rise to one electron being transmitted. For large vibrational coupling ( λ > ℏ ω ) a vibrational excitation will typically result in a few elec- trons being transmitted due to a non-vanishing elastic transmission. To illustrate a quantitative approach to ob- tain the parameters of a real system, we have used DFT to calculate coupling parameters for a benzene molecule interacting with two gold contacts. Since benzene is bound by van der Waals forces and only show a weak hybridization with metallic states it satisfies the mini- An explicit result for T$_{n}$ can be obtained by perfoming the integrals in Eq. (A1) after a Taylor expansion of e − $^{gf$_{τ,s,t}$}$. The result for the vibrational ground state T$_{0}$ involves L$_{0}$ ( x ) = 1 and has been calculated previously. 6 Here we simply state the result which is T$_{0}$ ( ε$_{i}$, ε$_{f}$ ) = Γ$_{L}$Γ$_{R}$ e − 2 g ∞ ∑ m =0 g m m ! δ ( ε$_{i}$ − ε$_{f}$ − m ℏ ω$_{0}$ ) ∣ m ∑ j =0 ( − 1) j ( m j ) ∞ ∑ l =0 g l l ! ( 1 ε$_{i}$ − ε$_{0}$ − ( j + l − g ) ℏ ω$_{0}$ + i Γ / 2 ) ∣ ∣$^{2}$ . (A4)