 due to their hybridization with the substrate, a partial occupation of the lowest unoccupied molecular orbital (LUMO) due to charge transfer from the substrate, and a substantial intramolecular Coulomb interaction $U=1.25$eV for the LUMO. The specific local environments of the two adsorption sites lead to slightly different positions of the on-top and bridge LUMOs, as can be seen in the projected densities of state in Fig. \[NRG\_fig4aa\], with the weight of the PDOS spectrum of the bridge molecule appearing further to the left. Similarly, the value of the hybridization functions at the chemical potential differ slightly for the two molecules, being $\Gamma^\mathrm{bridge}(0) = 190$meV and $\Gamma^\mathrm{top}(0) = 165$meV.

Because the Coulomb interaction is substantial compared to the hybridization strength, it enforces a single occupation of the LUMOs on both molecules, leading to a free spin of the radical that is ultimately screened by the Kondo effect for $T\to 0$. Since the *ab initio* PDOS spectra only contain the Coulomb interaction on a mean-field level, the half-filled orbitals are spin-degenerate in a paramagnetic calculation and the effective mean-field orbital energy must be pinned close to the Fermi energy, as is indeed apparent in Fig. \[NRG\_fig4aa\].

Both *ab initio* spectra in Fig. \[NRG\_fig4aa\] are much too wide compared to the STS data which exhibits a peak width of the order of $30$meV in the differential conductance at zero bias. This indicates that many-body correlations play an important role and must be taken into account for matching theory with experiment. Interpreting the *ab initio* PDOS as a mean-field solution [@PTCDAAgMove; @gated_wire_spectral; @AU-PTCDA-monomer; @AU-PTCDA-dimer] allows us to extract the single particle energies $\epsilon_{d\sigma}^\text{bridge}=-0.77\,\text{eV}$ and $\epsilon_{d\sigma}^\text{top}=-0.67\,\text{eV}$ for both types of molecules as well as the full complex hybridization function $\Delta(z)$.

![Comparison between the experimental $dI/dV(V)$ s