## 3 Results and Discussion
Results for all tip structures are presented in Table 2. As expected, the dependence with tip structure is opposite that of Table 1: the largest shift was found for the pyramid-pyramid combination, where the polarization self-energy was lowest, and structures with one pyramid followed. These differences are relevant for the calculation of conductance in subsequent sections. Furthermore, the total self-energy correction for the LUMO peak was 1.90 eV for the adatom-adatom, adatom-trimer and trimer-trimer tip structures, 2.09 eV for the adatom-pyramid and trimer-pyramid tip structures and 2.31 eV for the pyramid-pyramid tip structure. Having described the spread of the self-energy values with respect to tip structure, we applied the correction to the Hamiltonian (H\(\rightarrow\)H + \(\Sigma\)) as described in Equation (1). The shift was applied to the molecular Hamiltonian. DFT values are given by the eigenstates of the molecular box of the junction Hamiltonian, as described previously. The initial and final (corrected) energies of the HOMO resonance are given in Table 3. Figure 3 shows the corrected density of states projected onto the atoms of the molecule for tip structure combinations. The figure is centered on the range around \(-\)3.0 eV, near the position of the corrected HOMO peak. The inset reproduces the same data on a linear scale and over an extended energy range with the curves offset for clarity. The LUMO peak was shifted following the same procedure described here. The position of the corrected HOMO peak ranged between \(-\)2.9 and \(-\)3.2 eV. Additionally, the position of the corrected LUMO peak ranged between 4.5 and 4.8 eV. \begin{table}
\begin{tabular}{c c} \hline \hline  & \(\Sigma_{\text{HOMO}}\) (eV) \\ \hline Adatom–Adatom & \(-\)1.77 \\ Adatom–Trimer & \(-\)1.77 \\ Trimer–Trimer & \(-\)1.77 \\ Adatom–Pyramid & \(-\)1.96 \\ Trimer–Pyramid & \(-\)1.96 \\ Pyramid–Pyramid & \(-\)2.18 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Total self-energy correction (\(\Sigma_{\text{HOMO}}\) = \(\Sigma_{\text{HOMO}}^{1}\) + \(\Sigma_{\text{HOMO}}^{2}\)), calculated using the geometry of the molecule at the interface and a single image point charge. \begin{table}
\begin{tabular}{c c c} \hline \hline  & \multicolumn{3}{c}{**HOMO Peak Position (eV)**} \\ \hline  & DFT & DFT + \(\Sigma\) \\ \hline Adatom–Adatom & \(-\)0.84 & \(-\)3.04 \\ Adatom–Trimer & \(-\)0.83 & \(-\)2.91 \\ Trimer–Trimer & \(-\)1.09 & \(-\)2.93 \\ Adatom–Pyramid & \(-\)0.96 & \(-\)3.01 \\ Trimer–Pyramid & \(-\)0.97 & \(-\)2.98 \\ Pyramid–Pyramid & \(-\)0.93 & \(-\)3.17 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Highest occupied molecular orbital (HOMO) peak position, calculated using the DFT and DFT + \(\Sigma\) method. So far, we have focused on the DFT and corrected electronic properties, discussing how to calculate the magnitude of these corrections. In the final section of the paper, we turn to the electron transport properties and how to apply these corrections to DFT-based conductance calculations. Figure 4 shows the transmission spectra of the different BDA junctions calculated using the DFT-Landauer formalism. These calculations take the DFT-based electronic structure as input, with its well-known errors in resonance position. Figure 4 highlights the energy range below the Fermi level where the HOMO resonance, which defines zero-bias conductance, appears at the DFT level. As before, the inset plots the same data on a linear scale over an extended energy range. Figure 4 shows that low-bias conductance is determined by the tailing of the HOMO resonance into the Fermi level [25; 26; 51; 52]. In Figure 4, the HOMO peak position is found between \(-0.85\) and \(-1.00\) eV over the tip structures considered, obviously following the DFT-based density of states (see Figure 2 and Table 3). This is to be compared to the HOMO transmission peak between \(-0.94\) and \(-1.27\) eV reported in the literature [25; 26].