## 1 Introduction  
One of the main goals of molecular electronics is to mimic standard electronic circuits using molecules instead of p-n junctions like components based on silicon [1]. To do so, the first property to achieve in a molecular junction is to favor and understand at the fundamental level the circulation of the electronic current through the molecule. The problem of the electronic conduction mechanism in a molecule connected to metallic electrodes is very complex, and the possibility of many different regimes has been well presented theoretically by [PERSON] et al. [2; 3]. In particular, according to the molecular length, different regimes are observed, which exhibit different dependences in voltage and temperature. Hence, in the frame of elastic transport (considering that inelastic interactions only occur in the electrodes), for big molecular chains (longer than 5 nm), the electronic transport lies in an activated regime, called hopping regime, which is thermally activated. Indeed, in this regime, the conductance evolution of the molecular junction can be written as G \(\sim\) exp(-E\({}_{\text{A}}\)/k\({}_{\text{B}}\)T), where E\({}_{\text{A}}\) represents the hopping activation energy, around 0.5 eV, k\({}_{\text{B}}\) is the Boltzmann constant and T the temperature [4]. With respect to the molecular length, the conductance decreases linearly, which is easily understandable since the electrons have to jump (hop) from one molecular site to the nearest neighbor one [5; 6]. This behavior is characteristic of [PERSON]'s law, which we can observe at the macroscopic scale. For smaller molecular chains, (i.e., below 5 nm) [7], the regime is not activated anymore and corresponds to the direct tunneling of the electrons through the molecular junction, provided that the applied voltage is lower than the characteristic electronic barrier of the system. In this case, there is no temperature dependence, and it is a reasonable approximation to say that the current varies linearly with the voltage at low bias. Indeed, the current-voltage relation is often non-linear for a significant range of voltages, before the molecular level is brought into alignment with the Fermi level of the electrodes. The transition between both regimes has been observed around 4 nm in conjugated polymers [8]. Note that, for larger voltages, this regime is generalized to the Fowler-Nordheim tunneling, where the current varies like the inverse of the voltage [9]. A fundamental problem in electronic transport in molecules lies in the length dependence of the conductance in the molecular junction. This dependence is reflected in the so-called attenuation factor, which is representative of the electronic current propagation in the molecular junction. Since a molecular junction is generally not metallic, the electronic conductance decreases as a function of the molecular length. Obviously, bearing in mind that the main application of molecular electronics is to design new devices for future electronics, one is interested in the lowest possible attenuation factor in order to increase the current in the molecular circuit. In the direct tunneling regime that we will particularly consider here, the conductance decays exponentially with respect to the molecular length, as we will detail in the next section. The main goal is therefore to reduce this exponential decrease as much as possible in order to optimize the electronic flow in the molecular circuit. Hence, many experimental studies have been devoted to the determination of attenuation factors in different types of molecular junctions. In the meantime, theoretical methods have been developed to determine the electronic and transport properties of molecular junctions.