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My question is, in periodic real-space DFT, similar to what happens in VASP, is accurately determining the Fermi level (E_F) still challenging when compensating background charges are present? Or, in real-space DFT, is the calculation of the exact position of the Fermi level trivial?

I am not sure if I understand your question.
Are you referring to some feature in VASP when you say `when compensating background charges`?
Which feature?

As you know, in VASP, for charged systems, a uniform compensating background charge is automatically distributed (into the supercell) to maintain the charge neutrality of the system. However, this can lead to inaccuracies in the calculated Fermi level due to unphysical interactions arising from spurious charges. Articles like "Fermi Level Determination for Charged Systems via Recursive Density of States Integration" offer solutions to improve the accuracy of Fermi level shift calculations per injected electrons. Now, I want to ask whether in periodic real-space DFT, similar to what happens in VASP, we still encounter challenges in determining the exact Fermi level or not?

Yes, a uniform compensating background charge is added to maintain charge neutrality.
Which type of inaccuracies in the calculated Fermi level are you referring to?
I still don't understand your question, sorry.

Which physical property are you interested in calculating?

My question is quite clear, and I believe I have provided sufficient explanation. For some reasons, I would prefer not to explain more. Therefore, I would appreciate it if you could direct my question to someone who understands it. Thank you.