Accurate electronic and chemical properties of 3d transition metal oxides using a calculated linear response U and a DFT + U(V) method
Zhongnan Xu, Yogesh V. Joshi, Sumathy Raman and John R. Kitchin, “Accurate electronic and chemical properties of 3d transition metal oxides using a calculated linear response U and a DFT + U(V) method”, The Journal of Chemical Physics 142(14), 144701 (2015). https://doi.org/10.1063/1.4916823
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Question
Standard GGA functionals describe transition metal oxides poorly because the self-interaction error of localized d electrons does not cancel. The usual fix, adding a Hubbard U, is mostly done with U values fitted empirically to experimental data, which is not possible for properties that are hard to measure, such as adsorption on oxide surfaces. A U computed from first principles by the linear response method is system specific, but then each oxide has its own U, and total energies computed at different U values contain an arbitrary offset that makes reaction energies between them meaningless. The paper asks whether linear response U values can give both better band gaps and accurate reaction energies for bulk 3d oxides, and how to remove that offset (Xu et al. 2015).
Approach
All calculations used Quantum ESPRESSO with PBE and ultrasoft pseudopotentials. To show that conclusions do not depend on how the Hubbard projectors are defined, every calculation was done with two pseudopotential libraries, the original Quantum ESPRESSO set and the GBRV set, which give noticeably different linear response U values (for example 2.73 versus 4.86 eV for Cr2O3). Fourteen oxides of V, Cr, Mn, Fe and Co were studied, from rock-salt monoxides to V2O5, CrO3 and spinels, with U values roughly between 2.7 and 7.4 eV.
For reaction energies, the authors adapt the DFT + U(R) method of Kulik and Marzari, developed for molecular bond breaking, to bulk solids by replacing the bond length with the cell volume. The energy difference between two states is written as a path integral over volume and U. Along a path constrained to ground-state structures, the unphysical offset sits entirely in the term from the change of U, so only the pressure-like term (the volume derivative of the energy) is kept. They call this DFT + U(V). Each oxide’s energy is referenced, through a thermodynamic cycle, to isolated metal atoms and O2 at U = 0, so that oxidation energies become differences of DFT + U(V) cohesive energies. U = 0 was chosen for the atoms because their computed linear response U values were all near zero.
Key findings
With linear response U, band gaps agreed better with experiment for almost all oxides, where plain PBE gave gaps that were too small or absent. VO was worse and Mn3O4 barely improved, which the authors attribute to vacancy-free VO models and to the only available Mn3O4 measurement being on nanorods.
The full DFT + U(V) procedure was demonstrated on CoO by isotropically expanding the cell. The linear response U fell from the bulk value of 4.89 eV to about 1 eV, and integrating Hellmann-Feynman pressures along this U(V) path gave a reference energy lying between the cohesive energies at U = 0 and at the bulk U. Because this is expensive, they introduce a semi-empirical version in which the DFT + U(V) cohesive energy is a weighted average of those two cohesive energies, with weight x. For CoO the directly computed x was 0.600. The authors interpret x as a measure of how much of the d-electron self-interaction error is retained as bonds break.
Fitting x to oxidation reactions, the mean absolute error on a training set and a held-out test set was 0.139 and 0.157 eV per metal ion with the QE pseudopotentials, and 0.267 and 0.037 eV per metal with GBRV. (The text describes training on V, Cr and Fe reactions and testing on Mn and Co, while the figure caption lists different groupings.) Fitting a single x to all 14 reactions gave x = 0.599 ± 0.053 (QE) and 0.487 ± 0.08 (GBRV), reducing the error relative to plain PBE by 85% and 79%. The resulting errors were 0.14 (QE) and 0.20 (GBRV) eV per metal, compared with about 0.96 to 0.98 for PBE, 0.23 for empirical DFT + U and 0.40 for HSE06, the latter two taken from the literature. Formation energies computed the same way gave mixed results, reported only in the supplementary material, which was not available here.
Significance
The work shows that a first-principles U can replace fitted U values for both band gaps and bulk redox thermochemistry of 3d oxides, with a single near-universal weight (close to 0.5) and only a few DFT calculations per material. The agreement between the fitted and computed x for CoO supports the view that the calculated U, not the fit, drives the improvement. Referencing to atoms also avoids recomputing energies at globally or locally averaged U values, and it averages between the dissociated limit and the bulk U rather than between reactant and product U values. Because no experimental fitting of U is needed, the approach is aimed at properties such as oxide surface chemistry where reference data are scarce (Xu et al. 2015).