where ∆ d represents a continuous, ϵ-dependent coupling element, V ak , between the metal d-density of states and the adsorbate state. In line with Ref , it is taken to be a semi-ellipse centered around ϵ d with a width w d . ∆ 0 is a constant contribution applied to approximate the broadening of the adsorbate states into resonances upon interaction with sp-states of the metal, Λ is the Hilbert transform 14 of ∆ d +∆ 0 and ϵ a is the renormalised energy level of the adsorbate, after interaction with the sp states of the metal (see SI Section II for further details) and ϵ f is the Fermi level (set to 0). The factor of 2 in both terms represents degeneracy of the spin up and down states considered in the model. The adsorbate projected density of states is given by,