  We study the electronic coupling between an adsorbate and a metal surface by
calculating tunneling matrix elements H$_{\text{ad}}$ directly from first
principles. For this we employ a projection of the Kohn-Sham Hamiltonian upon a
diabatic basis using a version of the popular Projection-Operator Diabatization
approach. An appropriate integration of couplings over the Brillouin zone
allows the first calculation of a size-convergent Newns-Anderson chemisorption
function, a coupling-weighted density of states measuring the line broadening
of an adsorbate frontier state upon adsorption. This broadening corresponds to
the experimentally-observed lifetime of an electron in the state, which we
confirm for core-excited $\text{Ar}^{*}(2{p}_{3/2}^{-1}4s)$ atoms on a number
of transition metal (TM) surfaces. Yet, beyond just lifetimes, the
chemisorption function is highly interpretable and encodes rich information on
orbital phase interactions on the surface. The model thus captures and
elucidates key aspects of the electron transfer process. Finally, a
decomposition into angular momentum components reveals the hitherto unresolved
role of the hybridized $d$-character of the TM surface in the resonant electron
transfer, and elucidates the coupling of the adsorbate to the surface bands
over the entire energy scale.
