DFT + U and the Hubbard U
Local and semilocal exchange-correlation functionals (LDA and GGA) over-delocalize the d electrons of transition metal compounds because their self-interaction error does not cancel. Two common corrections are hybrid functionals, which mix in exact exchange, and DFT + U, which adds a Hubbard-like energy penalty on the occupations of localized d orbitals. The strength of that penalty, U, is most often fitted to experimental data such as oxide formation enthalpies, which works for known compounds but cannot be done where data are missing. Alternatively, U can be computed from first principles with the linear response method of Cococcioni and de Gironcoli, giving a system-specific value. The catch is that DFT + U adds a U-dependent shift to the total energy, so total energies of systems computed with different U values cannot be compared directly. See also Transition metal oxide thermochemistry.
Linear response U and the DFT + U(V) method (2015)
Xu, Joshi, Raman and Kitchin computed linear response U values for fourteen V, Cr, Mn, Fe and Co oxides with two pseudopotential libraries, finding that the values depend noticeably on the pseudopotential (for Cr2O3, 2.73 eV with the original Quantum ESPRESSO set versus 4.86 eV with GBRV). The calculated U improved band gaps over plain PBE for nearly all of the oxides. To compare energies at different U, they adapted Kulik and Marzari’s DFT + U(R) idea to solids: the energy change along a path through volume and U is split into a volume-derivative (pressure) term and a U-derivative term, and the latter, which carries the arbitrary offset, is dropped. For CoO, the U computed along an isotropic expansion fell from 4.89 eV to about 1 eV. A cheaper semi-empirical form writes the cohesive energy as a weighted average of values at U = 0 and at the bulk U, with weight x near 0.5 to 0.6 (Xu et al. 2015). Summary: xu2015c.
Linear response U for adsorption on rutile dioxides (2015)
Xu, Rossmeisl and Kitchin computed bulk linear response U values for ten rutile dioxides (from 3.32 eV for NbO2 to 7.15 eV for CrO2) and tracked OH, O and OOH adsorption on the (110) surface as U was raised from 0 to 8 eV. Apart from TiO2, any U > 0 made adsorption more endothermic while keeping the U = 0 scaling relations intact. For early 4d and 5d oxides the computed U sat where the adsorption energies stop varying smoothly with U, suggesting that large U is inappropriate for oxides with few d electrons, and for d0 TiO2 the bulk U appeared not to describe adsorption well, with DFT + U(R) suggested as a possible remedy (Xu et al. 2015). Summary: xu2015.
Sweeping U to find ordering changes in perovskite vacancy trends (2014)
Curnan and Kitchin proposed a two-step use of DFT + U for trend studies: first sweep a constant U over a whole series to find which neighbors change order, then compute first-principles U values only for those materials. For paramagnetic LaBO3 and SrBO3 vacancy formation energies, U on La 4f (0 to 12 eV) had no significant effect, while U on the B 3d states (0 to 8 eV) swapped LaVO3/LaCrO3 and LaMnO3/LaFeO3 between 3.5 and 4.0 eV, LaCoO3/LaNiO3 and SrCrO3/SrMnO3 between 4.0 and 4.5 eV, and SrCoO3/SrNiO3 near 6.0 eV. Linear response U values from Quantum ESPRESSO (6.2 to 10.7 eV for the La and Sr compounds tested) lay well above these thresholds. They also found that computed U shifted by up to 0.5 eV when the lattice constant changed by about 3.5% between GGA and GGA + U volumes, and noted that fitted U values from different pseudopotential sets can differ by up to 1.0 eV, so transferring U between codes is a significant assumption (Curnan 2014). Summary: curnan2014.