## 3 Theoretical Approaches  
In this section, we will discuss the most common theoretical approaches used to determine the electronic transport and in particular the attenuation factor in a molecular junction. An extensive review of electronic transport calculations using ab initio and density functional theory (DFT) based methods and Green functions can be found in [27]. Unfortunately, a direct relationship with the attenuation factor and overall a deep interpretation is not necessarily straightforward to deduce from these calculations. The objective here is to present the main contribution from the different parts of the molecular junctions that are characterized theoretically in each approach. Hence, the most common description of the electronic current in a molecular junction has been shown by [PERSON] [1; 28], considering the tunnel effect between metallic electrodes and a thin insulating film. In this respect, the current and the electronic conductance follow an exponential decay, as proposed in Equation (1), where the attenuation factor \(\beta\) can be expressed as \(\beta\sim\sqrt{(2\mathrm{m}\varphi/\hbar}\), with m the mass of the electron, \(\hbar\) the reduced Planck constant and where \(\varphi\) represents the electronic potential barrier of the system. In a molecular junction, this barrier is nothing other than the energy difference between the closest (non-resonant) molecular level to the Fermi level of the electrode and the Fermi level [1; 10]. Namely, if we consider electron transport, \(\beta\) will depend on the \(\mathrm{E_{LUMO}}\)-\(\mathrm{E_{Fermi}}\) difference, whereas, if we consider hole transport, \(\beta\) will depend on the \(\mathrm{E_{Fermi}}\)-\(\mathrm{E_{HOMO}}\) (\(\mathrm{E_{LUMO}}\), \(\mathrm{E_{HOMO}}\) and \(\mathrm{E_{Fermi}}\) being, respectively, the energy position of the LUMO, HOMO and Fermi level of the system). The immediate consequence of this interface property is that, in principle, beyond the nature of the molecular backbone, one could modulate the attenuation factor by changing the specific anchoring groups forming the bond between the molecule and the electrode. Indeed, the position of the molecular levels with respect to the Fermi level is mainly influenced by the anchoring groups used to connect the molecule to the electrodes. This aspect will be detailed later when considering properly the influence of the anchoring groups in well-established examples. However, this heuristic model is inspired by scanning tunneling microscope (STM) experiments, or the standard evolution of a wave function through a potential barrier in quantum mechanics, and represents the simplest expression of the Simmons model [28], currently used in molecular electronics to describe conductance attenuation in molecular systems. It works rather well for molecular junctions where the levels are close to the Fermi level but requires more ingredients when this is not the case. Other more sophisticated models highlight the role of the molecular backbones without taking into account the interfaces between the molecule and the electrode. For example, from an atomistic point of view, the conductance of the molecular wire can be understood using a simple tight-binding model, with N sites of energy \(\varepsilon_{1,}\)\(\varepsilon_{2\dots}\)\(\varepsilon_{\mathrm{N}}\) coupled through hoppings \(\mathrm{t_{1,2}}\), \(\mathrm{t_{2,3}}\), \(\dots\)\(\mathrm{t_{N\text{-}1,N}}\) and bridging the two electrodes [29] (see Figure 2). Notice that these energy sites are independent of the coupling at the interfaces, which automatically eliminates the role of the anchoring groups. Moreover, we can consider that all these energies are identical, namely to \(\varepsilon_{0}\), and the same for the hoppings, which are all equal to t. In this approach, the attenuation factor can be expressed as:  
\[\beta(\mathrm{E})=(2/a)\ln|(\mathrm{E}-\varepsilon_{0})/\mathrm{t}|, \tag{2}\]  
where \(a\) measures the segment size, for a total molecular length of N\(a\)[1].