# 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 '20 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 '20 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 '20 at 2:45