## 3 Theoretical Approaches
Considering the energy E as the Fermi level, and average values of \(|(\mathrm{E}-\varepsilon_{0})/\mathrm{t}|=10\) and a = 5 A, this expression gives typical values of \(\beta\) of around 9 nm\({}^{-1}\), independently of the anchoring groups or the electrodes. Note that a similar result can be deduced from the dispersion relation of an infinite linear chain:  
\[2\mathrm{t}\cosh(\kappa a)=\mathrm{E}_{\mathrm{Fermi}}-\varepsilon_{0}, \tag{3}\]  
with \(\kappa=\beta/2\), \((\mathrm{E}_{\mathrm{Fermi}}-\varepsilon_{0})/\mathrm{t}>>1\), and where the hyperbolic cosine stands for the exponential behavior at each side of the molecular chain where the electronic wavefunction tunnels from or to the electrode. In other words, this approach is also a generalization of the Simmons model. However, here, the energy difference \(\mathrm{E}_{\mathrm{Fermi}}-\varepsilon_{0}\) corresponds to the potential barrier of an infinite molecular chain without anchoring groups or electrodes, namely \(\varepsilon_{0}\) being an orbital of the infinite molecular backbone and not, for example, the HOMO level of the junction. Finally, the last important model lies in the determination of the attenuation factor through a complex bandstructure calculation [30; 31]. Similarly to what happens in solid state physics, an infinite molecular chain (analog to an infinite crystal) is considered and its bandstructure is calculated. Real wavevectors correspond to the usual molecular spectrum of the junction. The attenuation factor is determined from the calculation of evanescent states of the electronic wavefunction inside the molecule, namely the electronic states corresponding to complex wavevectors. These evanescent states, which correspond to the Shockley surface states in solid crystals [32], correspond to the different potential conduction channels of the molecular junction, originating from the different orbital hybridizations. Obviously, among all these different channels, the current will follow the channel with the lowest attenuation factor, similarly to the macroscopic behavior. Therefore, the value of the attenuation factor can be read on the complex bandstructure for the evanescent state with the smallest extension in the gap, as represented in Figure 3. This approach also leads to attenuation factors around 8 nm\({}^{-1}\) for alkane chains. Notice that an interesting mathematical derivation establishes a link between this result and a square root variation for the attenuation factor similar to  
Figure 2: Schematic representation of the bridge model with N sites of energy \(\varepsilon_{1}\), \(\varepsilon_{2\dots}\)\(\varepsilon_{\mathrm{N}}\) coupled through hoppings t\({}_{1,2}\), t\({}_{2,3}\), \(\dots\)\(t_{\mathrm{N-1,N}}\) and bridging the left and right electrodes. Reprinted figure from [1] and with permission. Copyright (2010) World Scientific. the one of the Simmons model. However, this expression is valid again only for the infinite molecule and not for the one whose HOMO and LUMO levels are determined by the coupling at the interface. To summarize the different theoretical approaches presented in this section, we can observe that each approach stresses a specific contribution. Hence, the model considers either the anchoring groups/electrodes (namely the coupling of the molecule at the interface) through the relative position of the molecular levels with respect to the Fermi level or the molecular backbone (namely the chemical nature of the molecular chain), in a rather exclusive manner. Consequently, it remains difficult to determine accurately and for all kinds of molecular junctions the corresponding attenuation factor. As such, a universal model does not seem to exist yet. Therefore, the debate remains open regarding whether the attenuation factor is determined by interface effects or only by the molecular backbone or by a combination of both. A large experimental consensus seems however to be established in favor of a major influence of the molecular backbone.