# How to calculate the potential in Hubbard-U correction and when to apply it?

I have seen that in certain cases Hubbard-U correction is relevant yet I have seen a number of papers that have not used it. So, I want to know if there is any theoretical basis for finding this potential or is it just by using empirical data. Also, what are the situations in which I have to use Hubbard-U potential.

I assume you mean including a Hubbard U potential within a Kohn-Sham density functional theory (DFT) calculation. The main reason why you might wish to do this is because you are studying a material for which you expect a significant self-interaction error.

The self-interaction error arises because the Hartree term in the DFT energy functional is the Coulomb interaction of the whole density with itself,

$$E_H = \frac{1}{2}\iiint{ \frac{\rho(\bf r)\rho(\bf r^\prime)}{4\pi\epsilon_0\left\vert {\bf r}-{\bf r^\prime}\right\vert} d^3{\bf r}d^3{\bf r^\prime}}.$$

This density has contributions from every particle, which means that there is a contribution to the energy associated with a particle interacting with itself - this is unphysical!

The "true" exchange-correlation functional would remove this spurious self-interaction and everything would be fine, but the approximate functionals can only remove the self-interaction approximately. If you look at the form of $$E_H$$ you can see that the self-interaction will be strongest when a particle's density is large, and since the total particle density must integrate to 1 this typically means the particle is strongly localised.

The usual cases where a Hubbard U should be considered are when modelling systems with d- or f-states, especially if they are partially filled. These states should usually be strongly localised, moderately low energy states (shallow core states), but because of the self-interaction their energy is raised spuriously and partially delocalised (because this reduces the self-interaction energy). It is not unusual for the energy of partially filled d- or f-states to be raised enough for them to approach the Fermi level, $$\mu$$, and hybridise with bonding states. There are other cases where these effects are significant (e.g. the N$$_2$$ triple-bond); the rule of thumb is that you need to worry about it whenever there are localised regions of high density arising from small particle numbers.

On a lighter note, I wrote a limerick to help people remember the key aspects:

I once had some d-states I knew
should be found at low E, not near $$\mu$$
this spurious reaction
to self-interaction
was cured with a small Hubbard U

For actually computing an appropriate Hubbard U, I refer you to the discussion here: What is the appropriate way of determining a value for the Hubbard-like potential U for LDA+U / GGA+U calculations?

NB using a Hubbard U is computationally efficient, but it is not the only way to solve this. One alternative method is to use a screened exchange functional, which explicitly removes self-interaction at short ranges (where the worst effects arise).

• +10. Nice one Phil! Nov 12, 2020 at 1:31
• If the materials contain some elements with strong correlation effects, can I include that with hybrid functional calculations, such as HSE06?
– Jack
Nov 12, 2020 at 2:37
• @Jack Hybrid functionals can reduce the self-interaction error, although I would encourage you to avoid anything with any unscreened Hartree-Fock component as it has a nasty discontinuity at the Fermi-level (screened Hartree-Fock is fine). I prefer to avoid the term "correlation" here, since it is often misused and defined differently in different fields. Self-interaction errors can occur whenever particles are strongly localised, it does not depend on them being correlated. Nov 12, 2020 at 2:45
• Dear Dr. @PhilHasnip thank you for this nice answer, as always. DFT+U is a method for treating electron over-delocalization due to the self-interaction error in the Hartree energy term. How come adding a +U repulsive potential localize electrons? Doesn't this repulsive +U potential further delocalize electrons by pushing them apart from each other?
– Sha
Sep 11, 2022 at 3:54
• @Sha good question! The easiest picture is probably to consider two degenerate states. The Hubbard U term favours integer filling of the states (one full, one empty); in contrast, self-interaction causes LDA (etc) to favour fractional filling of the states (both partially filled). The correct physical description is that the energy should be the same however we fill the states! If we add a Hubbard U term with the "right" strength, we can cancel the LDA self-interaction effect and recover the physically correct behaviour. Sep 21, 2022 at 23:17