You are mixing 2 concepts:

a) Yes, the final state rule applies and thus for XES calculations you should NOT use a core hole, but use the ground state DOS.

b) Strictly speaking, only E-tot is a valid quantity in DFT. In particular, the eigenvalues are in principle NOT excitation energies. However, for delocalized states (eg. valence electrons) experience has shown that the eigenvalues (bandstructure,DOS) can be used to interprete experimental spectra. This is, however, no true for localized states (core states), because these eigenvalues are not ionization energies but are defined as the derivative of the total energy with respect to occupation of the corresponding state.
     e_i =d E / d n_i
So one can use "Slaters transition state" approximation and make a supercell calculation, where on one atom HALF an electron should be removed. After scf, check the eigenvalue (with respect to EF) and you should obtain a significantly better binding energy of this core state (and the corresponding absolute XES transition energy. (Typically the error of a 1s core state is reduced from eg. 10 eV to about 1 eV. (about XPS energies and Slaters transition state see my lecture notes of our workshops on our web site).

Am 28.08.2015 um 17:19 schrieb Vladimir Timoshevskii:
Dear Wien2k users and developers,

I am working with experimentalists and try to simulate the XES measured
by soft x-ray detector, coupled with electron microscope. So, the
ionization source in this setup is the electronic gun of the TEM. The
test compound is hexagonal layered BN, which was quite well studied
before, including the similar setup. I would greatly appreciate if you
could share your opinion on the following 2 issues, which I am facing now:

i) Is it possible, in principle, to obtain correct photon energies
instead of shifting spectrum by hand to the EF position? I understand,
that the position of the core level (B-1s) is sensitive to the form of
the potential well, and the closer my potential is to the real one, the
better is the position of the core level. I tried different
XC-functionals, and found that actually the atomic-like Hartree-Fock
gives the best results: the whole spectrum (B K-edge) is shifted to
higher energies, closer to experiment, and the spectrum shape is also
much better. However, there is still ~10eV shift, relative to the
experimental spectrum. So, the XC-functional alone does not solve this
problem ...

ii) This is a more fundamental question, and is actually related to the
first one. I guess, the main reason for the photon energy
underestimation is the presence of the core hole, which shifts the
ionized core level to lower energies. I did several test calculations of
B-K spectrum using supercells of diffferent sizes with a core hole in B
1s. Indeed, by playing with fractional B 1s occupation (trying to catch
the "transition state"), it seems to be possible to shift the whole
spectrum to experimental position. But in this case, what about the
"rule of the final state"? According to this rule, the hole must be
created in the valence band (and screened out), and the core lavel must
be filled. This is what we normally assume ... Does that mean that the
XES calculations with hole in the core are unphysical, in spite of
giving better photon energies? May be, the situation here, especially
when we use TEM electronic gun for core ionization,  is more
complicated? In my opinion, the valence-core transitions are happening
in the potential, already distorted by the presence of the core hole. Am
I right? Then, how it agrees with the "rule of the final state"? Any
thoughts on that would be highly appreciated!

Thanks a lot in advance!

Vladimir Timoshevskii

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Peter Blaha
Inst.Materials Chemistry
TU Vienna
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