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

Question

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.

Answer 1

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).