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The Wiki states, with regards to using ICORELEVEL = 2 (explicit core hole calculation) that

[...] absolute energies are not meaningful, since VASP usually reports valence energies only. Only relative shifts of the core electron binding energies are relevant [...]

This is true in the context of how VASP reports these energies, i.e., as the difference between the all-electron total energy of the system and the summed energies of the PAW potentials (the "reference" energies). If these PAW reference energies were available, including the reference energy for the core-hole PAW potential (where the core hole state is spherically symmetric and the core is frozen), then one could obtain an estimate for the absolute binding energy of the core electron as the difference of total energies plus the differences between reference energies.

My question is: can these reference energies be retrieved somehow?

In general obtaining the absolute core energies is not very meaningful, as the DFT values are not accurate.

The absolute core electron binding energies can be computed following the method I mentioned with errors in the order of 1 or 2 eV, at least for light elements and 1s states. I can do this with GPAW, for instance, because I can retrieve the all electron energies from GPAW. So I suppose my question is how to retrieve them from VASP.

Could you be more specific which exact values you are looking for?

You are aware that a lot of information about the PAWs is printed in the header of the files, aren't you?

For the regular PAW potentials I can look at the POTCAR. For the core hole calculation (ICORELEVEL = 2), as far as I understand, the core hole PAW is generated dynamically during the calculation, and the all-electron reference energy is not printed in the OUTCAR. I want these reference energies, if they are accessible.

Can you check your OUTCAR file for "the core state eigenenergies are"? Are these the values that you need?

No, those seem to be the Kohn-Sham eigenvalues, whereas I need total energy values. I'll try to be more descriptive:

- When the PAW potential is generated by the VASP developers, an atomic calculation is done, which yields the total energy of the isolated atom E_0 (spherically symmetric valence orbitals w/o spin polarization). Let's say my isolated atom is C.
- When I run a "regular VASP calculation", say for diamond in a 64-atom supercell, this reference energy is (implicitly or explicitly, I don't know how VASP does it internally) subtracted from the all-electron total energy and reported by VASP as E. So E/64 is the "cohesive" energy per atom of diamond (minus the spin polarization effects, since E_0 does not include those, but let's forget about that). This is a small negative number, of the order of -10 eV per atom. This cohesive energy is the energy difference arising from (hypothetically) bringing all the individual neutral C atoms from very far away, to form the diamond lattice.
- The "all-electron cohesive" energy, which would be E_0 + E/64 per atom in this diamond example, is a much more negative number, something in the order of -1000 eV per atom for diamond. This is equivalent to computing the energy difference where the individual C nuclei C^{6+} and individual electrons (6 per C atom) are all initially separated by long distances and then brought together to form the diamond lattice.
- When I create a 1s core hole in one of the 64 C atoms in my diamond supercell, the E_0^* of that atom (and only that atom) is now different (denoted by *) and much higher in energy than E_0 (for C, something of the order of 200 eV higher).
- The energy now reported by VASP for this 63C + C^* system (where C^* indicates the excited core) is E^*. The all-electron total energy of the system is now E^* + 63*E_0 + E_0^*.
- The core-electron binding energy can be estimated somewhat accurately (much more accurately than using eigenvalues) by computing the difference between the final (excited core electron) and initial (regular ground state) all-electron total energies:

E_{BE} = E^* + 63*E_0 + E_0^* - (64*E_0 + E)
E_{BE} = (E^* - E) + (E_0^* - E_0)

The E^* - E total energy differences (i.e., the differences between the total energies reported by VASP) are not meaningful in absolute value, as correctly indicated in the documentation. However, the relative shifts when I change the atomic environments, for instance when I compute the binding energy for the graphite 1s core electron, are accurately computed if the atomic species (C here) does not change, because E_0^* - E_0 does not change. This is also correctly stated in the documentation.

The information I want to access, assuming that it can be retrieved, is E_0^* and E_0, because that allows me to estimate the absolute binding energies.

We currently do not calculate the total energy of the atomic problem with the core hole in VASP. If you want to try to implement the functionality yourself, you can inspect the routines in rhfatm.F, which calculates the eigenvalue problem. Essentially what would need to be done is evaluating the expectation value of the resulting wave functions to get the Hartree and the xc energy.