How are the energies of, e.g. the 4p orbitals of the first transition series, determined?

Take a first row transition metal, say manganese. The 4s and 3d orbitals are occupied in Mn; each orbital's energy can can be definitely determined by ionising that orbital, obtaining the resulting Mn+ ion's energy, and comparing with the energy of the neutral atom. This can easily be done either empirically or in silico; there's no problem up to this point.


However, a problem occurs when one tries to obtain the energy of the 4p orbital. This orbital is not occupied in the ground state configuration of Mn, so ionisation is not possible here.


There seem to be two ways to solve this problem:


* One by exciting, say, a 3d electron into a 4p shell and comparing the resulting (excited-state) Mn's energy with that of a 3d-ionised manganese cation(i.e. the "band-gap" method).
* The other by pushing an electron into a 4p orbital of an Mn atom, creating an anion, and comparing its energy with that of a neutral Mn atom(i.e the "electron affinity" method.


But the two methods are not consistent with each other since the LUMO, and hence the band gap, of a chemical species is in general NOT correlated to its electron affinity.


There seem to be some data on the 4p orbital energies of first row transition metals, e.g. [over here](https://www.google.com/url?sa=t&source=web&rct=j&url=https://www.colby.edu/chemistry/PChem/notes/AOIE.pdf&ved=2ahUKEwiD5q2BoKT2AhU_sFYBHQIMBAcQFnoECEAQAQ&usg=AOvVaw3I_QdHI3xbLOEdq0WmYnpf); which of the two (non-consistent) methods I mentioned was/is used in the literature to determine the 4p orbital energies of first row transition metals?


UPDATE: After further personal research on this issue, I found out on Wikipedia that the Kohn-Sham DFT theory exactly captures both the excitation energy("first way"), resp. the ionisation energy("second way"), by simply taking them to be the difference between the filled and unfilled orbitals' energy levels("first way"), resp. the filled orbital energy levels("second way").


So I'm going to re-ask it: are the "energy levels" of unoccupied valence orbitals of certain atoms, say transition metal's np orbitals, known in the literature derived from Kohn-Sham DFT theory, [using the known (or calculatable) exact electron densities](https://www.nature.com/articles/s41467-019-12467-0) for these atoms?


UPDATE 2: The link basically says that, (even though the general functional is still unknown) the functional for a **specific** species can be derived exactly once the CI(configuration interaction) density for that species is known. And CI calculations are relatively straightforward for bare atoms.


