## 3 Results and Discussion  
We began our analysis by studying the electronic properties of the junctions within DFT, as well as their variation with respect to the binding motifs (Figure 1). We focused only on the position of the HOMO peak, as it has been shown in the literature that HOMO dominates the zero-bias conductance for BDA [25; 26; 51; 52]. Figure 2 shows the density of states projected over the atoms of the molecule for all the binding motifs considered: adatom-adatom, adatom-trimer, trimer-trimer, adatom-pyramid, trimer-pyramid and pyramid-pyramid. The figure is centered around the energy of the DFT HOMO peak (\(-1.0\) eV). The inset shows the same data on an extended energy range. The data are offset vertically for clarity. From the figure, the position of the HOMO peak at the DFT level had a small variation across all different structures, ranging from \(-0.83\) to \(-1.09\) eV. The LUMO peak was found between \(2.74\) and \(2.92\) eV. From Figure 2, tip structures involving adatoms broadly resulted in sharper PDOS peaks and resonances closer to the Fermi level. In order to implement the DFT + \(\Sigma\) method, it is necessary to calculate the magnitude of the self-energy correction of the appropriate resonance (Equations (2) and (3)). We first address the calculation of the gas-phase self-energy correction of the HOMO peak, \(\Sigma_{\rm HOMO}^{1}\), which involves the calculation of the isolated molecule. Several ways of computing this correction are possible. One option is to optimize the geometry of the molecule. Another possibility is to compute the molecule in the geometry it adopts at the junction. In the case of BDA, the magnitude of both corrections was the same in all cases regardless of which approach was used, \(-2.99\) eV, although we believe that this is due to the reduced conformational flexibility of the amine linker. We anticipated a spread of values for other linkers, such as methyl-sulfide groups, which rotate when adsorbed with respect to their gas-phase geometry. In this case, it would be more consistent to use the geometry the molecule adopts at the junction [20; 35]. The second term in the self-energy correction, accounting for the polarization due to the metallic surface, \(\Sigma^{2}_{\text{HOMO}}\), was approximated using a classical image charge model, as described in Equation (4). A single point charge was positioned at the geometrical center of the molecule [28; 29; 53]. The image plane position was taken to be 1.0 A above the outer atomic plane of each Au [62]. Results for all tip structures are presented in Table 1. The calculated shift of the HOMO resonance due to screening is, for both approaches, given in Table 1. We see that the largest values were obtained for \"short\" tips that protrude the least from the surface, such as adatoms or trimers. When pyramidal tips were considered, image charge screening was reduced due to the larger vertical distance spanned by these tips. \begin{table}
\begin{tabular}{c c} \hline \hline  & \(\Sigma^{2}_{\text{HOMO}}\) **(eV)** \\ \hline Adatom–Adatom & 1.22 \\ Adatom–Trimer & 1.22 \\ Trimer–Trimer & 1.22 \\ Adatom–Pyramid & 1.03 \\ Trimer–Pyramid & 1.03 \\ Pyramid–Pyramid & 0.81 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Polarization due to the metallic electrodes, \(\Sigma^{2}_{\text{HOMO}}\), calculated using a classical image charge model with a single point charge. Figure 2: Density functional theory (DFT) density of states projected over molecular atoms for BDA junctions with different tip structures: adatom–adatom, adatom–trimer–trimer, adatom–pyramid, trimer–pyramid and pyramid–pyramid. Data are vertically offset for clarity. The plot is centered on the position of the HOMO peak (\(\leftarrow\)\(-\)1 eV). The inset shows the same data on a wider energy range. The total self-energy correction for the HOMO peak is the sum of both (opposing) contributions.