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I often see DFT+U corrections for transition metals with delocalization problems, but a comment about applying a U correction to nitrogen or oxygen's p orbitals also shows up when digging around in the literature. It seems this is not done often, which makes understanding why and when it is done more difficult to search.

When is this appropriate to consider?

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I've done some work with applying U to oxygen p states. I've seen some vague arguments against it, which usually amount to "they're not as localized as d states", which of course can be true (d states can also be quite itinerant, like $\sigma^*$ states in some metallic nickelates). The real question though, is how "localized" does a state need to be before a U correction offers some improvement in the description of the material. No one seems to have an answer for that.

So instead of making hand-wavy intuition arguments, I'd rather let the proof be in the pudding. There is no reason a priori to assume that you can never apply U to oxygen p states in some circumstances. It might not be done often in the literature because so much of the literature is using empirical (or not even empirical, "second-hand" empirical from other papers that could have been using completely different parameters!) U values that make it more complicated to apply U to more states. Self-consistent U is very helpful here.

In my last paper (shameless plug) I devoted a small section (section IV.C.7) to discussing this. In that work I used the ACBN0 method to calculate values of U for d and p states on perovskite oxides. I also applied a "naive" U from literature, only to the d states. The character of the bonding can be very different between the two. Typically, applying U to both d and p gave electronic structures that are in better agreement with HSE hybrid functional calculations, as well as almost universally better agreement with experimental lattice geometry. In ZnO, it's often discussed in the literature that you actually need to apply U to oxygen in order to increase the (KS) band gap to experimental values while keeping the magnitudes of U values physically realistic. In addition, when interpreting experimental XPS, XES and XAS spectra in the Zaanen-Sawatzky-Allen framework, there are significant Coulombic U interactions between p orbitals in many oxides. Often the magnitudes are larger than the d states because the orbitals are more populated. So there is at least some experimental support for this being a reasonable correction.

That being said, you can also get better results beyond applying U to only d states in other ways, such as using DFT+U+V. But that method also doesn't preclude applying corrections to p states either (in fact V in many oxides will be a d-p interaction).

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    $\begingroup$ @Anyon In addition to ACBN0 (which is ab initio and self-consistent), ab initio values of $U$ have been calculated for oxygen $p$ states with cRPA, (10.1103/PhysRevB.87.165118, 10.1103/PhysRevB.86.165105, 10.1088/1367-2630/16/3/033009, 10.1103/PhysRevB.86.165124) and also the recent DFPT method, which is equivalent to linear response. I can't find those papers right now though... $\endgroup$ Oct 20 '20 at 7:38
  • $\begingroup$ Thanks for the references. I removed my initial comment after recalling I had seen it done, but I didn't know where. It may have been that Biermann paper. $\endgroup$
    – Anyon
    Oct 22 '20 at 21:53
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I would like to take a poke at this even though my answer might not be completely satisfactory.

The '+U' correction to DFT methods is typically useful for systems with highly localized orbitals - For example, the 3d orbitals of transition metals. But what's also interesting is that typically one also associates a Hubbard U value to the Oxygen/Nitrogen 3p orbitals in a calculation, if they are present in the material. This is what you have mentioned in the question. The idea is simple - you want to include a value of Hubbard 'U' so that these highly localized states are pushed away from the fermi level, otherwise they might hybridize incorrectly. From what I've seen, from systems like NiO, TiO etc which are very well researched, a value of 'U' is also applied for Oxygen. In these systems, the most localized orbitals are 3d of the Transition metal and 3p of the Oxygen. I might imagine that it is always better to use a value of 'U' for O/N - the papers that assume a value of zero have probably concluded that inclusion of a non-zero value does not quantitatively affect the results.

I would love to hear more if anyone else has insights on this.

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    $\begingroup$ "the papers that assume a value of zero have probably concluded that inclusion of a non-zero value does not quantitatively affect the results" I admire your optimistic point of view :) $\endgroup$ Oct 19 '20 at 16:16

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