content_image/420127.jpg content_image/420141.jpg content_image/420139.jpg content_image/420142.jpg content_image/420140.jpg content_image/420138.jpg
more, because of the uncertainty of the exact geometry of the single-molecule junction in the experiments, there is a corresponding uncertainty about how to model the coupling between the molecule and the electrodes and correspondly for the potential drops at each molecule- electrode contacts. Hence we take the quantitites char- acterizing the potential drops at the contacts as phe- nomenological parameters $^{24,61}$. 

We concentrate in this paper on the electronic prop- erties of the molecular junction in both equilibrium and non-equilibrium conditions as well as on the linear- response properties of the junction (prior to considering the full non-equilibrium transport properties in a forth- coming paper). Such properties are given by the density of electronic states and represented by the spectral func- tions, which are at the very heart of all physical proper- ties of the system, such as the charge density, the current density, the total energy, etc. 

Spectral functions are most closely related to photoe- mission and adsorption spectroscopies. To our knowl- edge such experiments have not yet been performed on single-molecule junctions, though photoemission spectra have been measured on quasi one-dimensional supported atomic scale metallic wires (see for example Ref.[62] showing interesting results on one-dimensional collective electronic excitations). 

The paper is structured as follows. We start with a description of our model system in Section II A and a discussion of the relevant underlying theory of non- equilibrium Green’s functions in section II B. Our cal- culated spectral functions are presented in Section III, where we consider first the equilibrium case (Section III A) and then the non-equilibrium case (Section III B) at the Hartree-Fock level. We discuss especially the ef- fects of including or not the Hartree diagram in the cal- culation. We also compare NEGF calculations with re- sults obtained from inelastic scattering techniques 16,63,64 for equivalent model systems in Section III C. We show that it is indeed necessary to go beyond SCBA to obtain correct results for the relevant range of electron-vibron coupling. We then present the effects of the second- order diagrams in the spectral functions in Section III D. The second-order diagrams correspond to two classes of process; the first is related to vertex corrections of the SCBA calculation and the second to polarisation effects (i.e. partial dressing of the vibron propagator by the electron-hole bubble diagram). In Section III E we dis- cuss the effects of different levels of approximation for the electron-vibron coupling on the linear conductance of the molecular junctions. Throughout the paper, we will use the term vibron to define a quantum of vibration of a mechanical degree of freedom. 

<img src='content_image/420127.jpg'>

Our model is based on a system with an interact- ing central region connected to two non-interacting elec- trodes (see figure 1). 

The total Hamiltonian for the system is given by 

where H$_{e}$,H$_{vib}$ and H$_{e}$$_{-vib}$ represent the electronic, vi- bron, and electron-vibron coupling parts of the Hamil- tonian respectively. The electronic part of the Hamil- tonian is broken into sections describing the left (right) electrode H$_{L}$ ( H$_{R}$ ), the central interacting part H e C and the potentials coupling the central region to the left and right electrodes V$_{LC}$ + V$_{CR}$ respectively: 

The Hamiltonians for the electrodes are given by 

where c † α ( c$_{α}$ ) creates (annihilates) a non-interacting electron with energy ε on electrode α . The electronic Hamiltonian for the central region and the coupling po- tentials are given by 

<img src='content_image/420141.jpg'>

<img src='content_image/420139.jpg'>

## II. MODEL 

## A. Hamiltonian 

<img src='content_image/420142.jpg'>

<img src='content_image/420140.jpg'>

<img src='content_image/420138.jpg'>

