DFT + U and the Hubbard U

A research topic across the group’s papers.

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.

Linear response U and the U window for TiO2 polymorph ordering (2015)

Curnan and Kitchin mapped the range of Ti 3d U over which DFT+U gives the experimental TiO2 polymorph ordering and compared it with linear response U values computed in VASP. With PBE and standard PAW pseudopotentials the window was about 2.8 to 4.3 eV and the computed U values for rutile, anatase, brookite and columbite (2.93 to 3.10 eV, uncertainty about 0.13 eV) all fell inside it; a U near 3 eV gave a rutile-anatase difference of 0.00 to 0.01 eV per TiO2. Semicore Tipv and Tisv pseudopotentials raised both the computed U (4.77 and 6.03 eV for rutile) and the window (4.7 to 7.0 and 5.8 to 8.2 eV), showing that U values are tied to the pseudopotential and are not transferable between setups. PBEsol lowered U but its window narrowed to 1.5 to 2.1 eV, so the joint shift did not carry over to a change of functional (Curnan 2015). Summary: curnan2015.

Papers on this topic