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TLDR: DFT+U is a cheap scheme to help localize electrons on atomic orbitals by penalizing fractional occupancies of such orbitals. However, when one tries to localize an electron/hole on an atomic complex such as a dimer/trimer, this will fail badly since the electron/hole needs to be shared between several atoms (leading to fractional occupancies which DFT+U actively punishes instead of favorizing them). Is there a generalization of the DFT+U scheme that applies to localizing electrons on orbitals more complex than atomic orbitals?

The DFT+U scheme for localizing electronic states:

Plain semilocal DFT (e.g. based on GGA or LDA) suffers from the infamous self-interaction error which tends to excessively delocalize electrons and unphysically favor metallic states.

Many schemes have been proposed to correct for this. Hybrid functionals have been proven successful (providing correct tuning of the exact exchange fraction), but expensive. Another handy solution has been DFT+U, based on the Hubbard model from the tight-binding picture. In Dudarev's formulation, this amounts to adding a correction term to the total energy roughly of the form: $$\Delta E_{DFT+U}=\frac{U_{eff}}{2}\sum_\sigma (Tr(n^\sigma-n^\sigma n^\sigma) )$$ Where $n^\sigma$ is the occupation matrix for spin $\sigma$ of the atomic orbital we applied the Hubbard term to. Qualitatively, we see that this term can be thought of as a penalty term that will be minimized in the case of idempotency $n^\sigma = n^\sigma n^\sigma$. This corresponds to the case of an occupation matrix that will have integer eigenvalues (0 or 1, fully unoccupied or occupied)

Long story short, this means that DFT+U will penalize fractional occupancies of atomic orbitals and favor fully occupied or unoccupied states, helping to localize electrons on atomic orbitals. But what if I want to localize electrons (or holes), just not on atomic orbitals?

The problem: localization on molecular-like orbitals

In my case, I am interested in modeling hole trapping on an $I_2^-$ dimer. The problem is the following: localizing a single hole on a dimer means splitting the hole 50/50 between the orbitals of each I ion, meaning half occupancies on both orbitals, which is actively disfavored by DFT+U!

Sure enough, when experimenting with DFT+U, I do observe that higher U values actually go against hole localization in the case of a dimer, and we find ourselves in an awkward situation where DFT+U now promotes delocalized states in this configuration. Bad DFT+U.

A tempting fix would be to use negative U values to favor half occupancies (or half vacancies in the case of a hole) of atomic orbitals on the dimer. However, this seems hard to justify physically in the context of the Hubbard model and seems too ad-hoc to yield meaningful results.

Another alternative would be to have more flexibility in the formulation of the Hubbard term, and instead of using the occupancy matrices of atomic orbitals $n^\sigma$, formulating the penalty term in terms of some form of "molecular orbital occupancy" of the dimer $n_{I_2^-}^\sigma$, so that the hole localization on a dimer orbital could be favored.

Has this kind of thing been studied/developped before? Does anybody know of a DFT+U formulation that allows for more general localization schemes that are not necessarily limited to a single-atom basis?

This more of a general question, but anything pertaining to a possible quantum-espresso implementation would be appreciated too

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    $\begingroup$ I think it's generally known as "DFT+U+V", where V is the "off-diagonal" element of the multi-element Hubbard potential. It's important in N2, for example, where the triple bond suffers from strong self-interaction error with many semilocal XC functionals. I don't know much more than this, but I can try to write an answer if none of the experts do! $\endgroup$ Nov 20, 2023 at 10:19
  • $\begingroup$ Thanks a lot, @PhilHasnip , and sorry I forgot to answer you. This was indeed exactly what I was looking for. I will let you turn your comment into an answer if you wish to, otherwise I'll answer my own question with a few additional details and useful links. $\endgroup$ Feb 7 at 9:05

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