How to calculate formation energy using DFT+U?

Question

I noticed that a lot of articles use DFT+U methods to calculate different formation energies when using regular DFT and DFT+U methods. For example, in the attached paper[1] on $\ce{In2O3}$, the enthalpy of formation was found to be $\pu{-9.86 eV}$ and $\pu{-10.41 eV}$ using LDA and LDA+U, respectively.

When calculating the formation energies using DFT+U, I am wondering if the energies of the elemental reference states ($\ce{In}$ and $\ce{O2}$ in the above case) also need to be calculated using DFT+U, or just from regular DFT, since they often do not have band gaps that DFT+U seeks to correct.

1. Pakpoom Reunchan, Xin Zhou, Sukit Limpijumnong, Anderson Janotti, Chris G. Van de Walle, Vacancy defects in indium oxide: An ab-initio study, Current Applied Physics, Volume 11, Issue 3, Supplement, 2011, Pages S296-S300,DOI: 10.1016/j.cap.2011.03.051

Answer 1

In the paper mentioned in the question, the $+U$-correction is applied to the semi-core $4d$ states of indium to increase the band gap of In₂O₃, which is underestimated by DFT. The band gap is important here, because the focus of the paper are charged defect formation energies, and for those the valence and conduction band edge energies are important. Otherwise, it is not too common to apply $+U$-corrections to completely filled shells.

Yes, generally, for the calculation of the formation energy of a compound, the constituents must be calculated using the same theory as the compound. In the example of the paper, bulk indium must then also be treated with LDA$+U$. There are no $U-$corrections applied to oxygen in this example, so the oxygen reference will simply be computed using LDA.

DFT is relatively good at describing delocalized metallic states, and applying $+U$-corrections to metals is problematic (the density of states is adversely affected: reduction near the Fermi level, e.g.). Therefore, as mentioned by @Tristan Maxson, at the materials project, metallic phases are computed with DFT, and transition metal oxides with partially filled $d$-shells and self-interaction-error problems with DFT$+U$. Oxide formation energies are calculated by adding a constant correction energy per $U$-corrected ion, which was fitted against a number of experimental heats of formation. This requires a constant $+U$ value for a given transition metal ion. Optimal $+U$ values will however depend on the coordination of the transition metal ion, its oxidation state, etc., and the materials project correction scheme (Jain et al., PRB 84, 045115 (2011)) has been extended to oxide-specific $U$-values (Aykol and Wolverton, PRB 90, 115105 (2014) and Voss, J. Phys. Commun. 6 035009 (2022)). With these methods one can thus compute formation energies by combining DFT and DFT$+U$ calculations, with $U$-values appropriate for a given system (even $U=0$ for metallic systems).

Answer 2

As far as my knowledge, they must be calculated using DFT+U. When you calculate the energy of $\ce{In}$ in its bulk state using DFT, it won't lead to the same value using DFT+U.

Answer 3

To calculate formation energies from DFT+U, you must use a self consistent set of parameters. This is easy to visualize since you can imagine the reaction of InO_DFT -> InO_DFT+U, which will end up having some non-zero reaction energy. While this example is obviously incorrect, mixing energy differences with DFT+U corrections and non-corrected differences results in a similar picture, but is often less obvious.

There have been some attempts to mix these however and materials project has a small section on a mixing correction they have applied, however I do not know personally if this has been successful or not. Someone more familiar with materials project might know more.