## 3 Results and Discussion
Calculated conductance values, given by the transmission at the Fermi level, are given in Table 4. Furthermore, as is common for DFT-based approaches, calculated conductance significantly exceeds the experimental value (\(6.4\times 10^{-3}\) G\({}_{0}\)) [25; 51; 52], in this case, by about a factor 10. Figure 3: DFT + \(\Sigma\) density of states projected onto molecular atoms for BDA junctions with different tip structures. Data are offset vertically for clarity. The inset shows the same data over a wider energy range. To improve the agreement between calculated and measured conductance, it is necessary to correct the position of the conducting orbital. The implementation of DFT + \(\Sigma\) in a fully self-consistent DFT-NEGF cycle, out of equilibrium, is far from simple. However, trends can often be drawn from studies in equilibrium. Since in many molecular junctions, conductance takes place due to non-resonant tunneling [3; 11; 34], it is illustrative to consider simple models involving only one molecular resonance. Although single level models fail when transport involves several molecular orbitals [63] or in cases of quantum interference [64], they nevertheless provide a good starting point for many representative molecular junctions. Different ways of fitting the relevant parameters and their accuracy have been discussed [9; 35; 44; 59; 63; 64; 65; 66; 67]. Here, we consider a single level (HOMO) at zero bias. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Conductance (\(10^{-2}\) G\({}_{0}\))**} \\ \hline  & DFT-Landauer & Lorentzian (DFT) & Lorentzian (DFT + \(\Sigma\)) \\ \hline Adatom–Adatom & 6.61 (10.33\(\times\)) & 6.66 (10.40\(\times\)) & 1.00 (1.56\(\times\)) \\ Adatom–Trimer & 6.07 (9.48\(\times\)) & 6.64 (10.38\(\times\)) & 1.03 (1.61\(\times\)) \\ Trimer–Trimer & 5.55 (8.68\(\times\)) & 6.25 (9.77\(\times\)) & 1.08 (1.69\(\times\)) \\ Adatom–Pyramid & 6.66 (10.41\(\times\)) & 5.66 (8.85\(\times\)) & 0.64 (1.00\(\times\)) \\ Trimer–Pyramid & 6.33 (9.88\(\times\)) & 4.62 (7.21\(\times\)) & 0.63 (1.00\(\times\)) \\ Pyramid–Pyramid & 8.10 (12.65\(\times\)) & 3.23 (5.05\(\times\)) & 0.35 (1.82\(\times\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Conductance for BDA junctions using different tip structures from DFT-Landauer and from Lorentzian fits to DFT HOMO or to DFT + \(\Sigma\) HOMO. Numbers in parenthesis indicate the overestimation of conductance compared to the measured value (\(6.4\times 10^{-3}\) G\({}_{0}\)) [25; 51; 52]. Figure 4: DFT-Landauer transmission spectra of BDA junctions using different tip structures. The figure highlights the tail of the HOMO resonance (\(\sim\)\(-1\) eV at the DFT level) into the Fermi level. The inset shows the same data over an extended energy range. First, the DFT-based conducting peak (in this case HOMO) is fitted using a Lorentzian model of the form,  
\[T(E)=\frac{A}{\left(\left(E\ -\ E_{\text{HOMO}}\right)^{2}\ +\ \left(\Gamma/2 \right)^{2}\right)}, \tag{5}\]  
where \(E_{\text{HOMO}}\) and \(A\) are the HOMO peak position and amplitude, respectively, and \(\Gamma\) is the full width at half maximum. The position of the resonance is then corrected using the self-energy previously addressed (\(\Sigma_{\text{HOMO}}\)). This model assumes that width of the Lorentzian (\(\Gamma\)) is unchanged, which is reasonable since DFT captures the electronic coupling well. For the BDA junctions considered, \(\Gamma\) takes three range of values. For adatom-adatom, adatom-trimer and trimer-trimer structures, the widths were 0.51, 0.52, and 0.55 eV, respectively. For adatom-pyramid and trimer-pyramid structures, the widths were 0.43 and 0.45 eV. Finally, for the pyramid-pyramid structure, the width was the lowest, 0.35 eV. This agrees well with previous calculations, where the conductance peak width showed a modest variation between 0.34 to 0.56 eV [25].