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Investigating the Energetic Ordering of Stable and Metastable TiO2 Polymorphs Using DFT+U and Hybrid Functionals

The Journal of Physical Chemistry C (2015)

Matthew T. Curnan and John R. Kitchin, “Investigating the Energetic Ordering of Stable and Metastable TiO2 Polymorphs Using DFT+U and Hybrid Functionals”, The Journal of Physical Chemistry C 119(36), 21060-21071 (2015). https://doi.org/10.1021/acs.jpcc.5b05338

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Question

Experimentally, rutile is the only stable bulk TiO2 polymorph, anatase and brookite are metastable, and columbite (the alpha-PbO2 type) is the next phase reached under pressure. Plain LDA and GGA calculations, however, often place anatase below rutile, and the answer changes with the functional. The authors asked whether a Hubbard U on the Ti 3d states, or a tuned fraction of exact exchange in a hybrid functional, can recover the experimental ordering consistently across functionals and pseudopotentials, and whether a first-principles (linear response) U lands inside the window where the ordering is right (Curnan 2015).

Approach

Rutile, anatase, brookite and columbite TiO2 were computed with VASP and PAW pseudopotentials, with ground-state energies taken from Birch-Murnaghan equation of state fits. The Dudarev form of DFT+U was applied to Ti 3d only (no U on O 2p), scanning U from 0 to 6 or 9 eV in 1 eV steps. Functionals were LDA, PBE, PBEsol, AM05 and PW91, and with PBE five Ti/O pseudopotential combinations were tested (standard Ti, Ti_pv and Ti_sv semicore variants, with standard or soft O). Linear response U values were computed in 48 to 96 atom supercells. Separately, rutile and anatase were computed with PBE0 and HSE06 while varying the exact exchange fraction from 0 to 1.

Key findings

With PBE and standard Ti plus standard or soft O pseudopotentials, the experimental ordering (rutile < anatase < brookite < columbite) held for U between roughly 2.8 and 4.3 eV. PW91 and AM05 gave similar windows (the combined GGA window was about 2.8 to 4.0 eV). Semicore pseudopotentials shifted the window to higher U, about 4.7 to 7.0 eV for Ti_pv and 5.8 to 8.2 eV for Ti_sv. PBEsol gave only a narrow window of about 1.5 to 2.1 eV because the anatase-rutile energy curve shifted upward, and LDA never placed columbite as least stable at any single U, so it never reproduced the full ordering. The authors connect the constant LDA and PBEsol shift to smaller equilibrium volumes rather than electronic structure.

Linear response U values with PBE and standard pseudopotentials were 3.10 (rutile), 2.93 (anatase), 2.98 (columbite) and 2.93 eV (brookite), each with an uncertainty near 0.13 eV, so all four fall inside the 2.8 to 4.3 eV window and share a common U range for direct comparison. Semicore pseudopotentials raised the computed U (rutile 4.77 eV with Ti_pv and 6.03 eV with Ti_sv), roughly tracking the shift in the window for rutile but not fully for anatase. PBEsol lowered U (2.73 and 2.56 eV for rutile and anatase), which missed its narrow window. The Ti_pv rutile value agreed within uncertainty with a published Quantum ESPRESSO value of 4.95 eV using a semicore GBRV pseudopotential. Using U near 3 eV gave a rutile-anatase energy difference between 0.00 and 0.01 eV per TiO2, consistent with experiment.

For hybrids, both PBE0 and HSE06 start at an anatase-favoring difference of about -0.081 eV/TiO2 and rise with exact exchange, crossing to rutile-favoring at about 0.72 (PBE0) and 0.78 (HSE06). The trend is not monotonic: it peaks (PBE0 near 0.040 to 0.045 eV/TiO2, about 0.015 eV above HSE06) and reverts to anatase-favoring toward pure Hartree-Fock exchange. The non-monotonic behavior tracked cell volume and the c/a ratios of rutile and anatase, suggesting structure (and possibly missing short-range dispersion) matters beyond 3d localization. Errors from an arbitrary exchange fraction reached 0.068 to 0.127 eV/TiO2, comparable to the 0.1 to 0.2 eV window relevant to epitaxial stabilization.

Significance

The study shows that DFT+U with a linear response U is a non-empirical route to experimentally consistent TiO2 polymorph energetics with GGA functionals, while hybrid functionals have no first-principles way to choose the exchange fraction and are therefore questionable as a predictive tool for BO2 polymorph selection. It also documents that pseudopotential choice shifts both the computed U and the U window together, a practical caution for comparing U values across codes. The authors note that results are internal energies; vibrational and entropic terms are discussed but not computed (Curnan 2015).

Related

Concepts: Oxide polymorphs, DFT + U and the Hubbard U, Exchange-correlation functionals.

Builds on the functional-dependent BO2 polymorph energetics of mehta2014 and relates to the TiO2 epitaxy study xu2017. The cited linear response U for rutile dioxides is from xu2015; a companion study by the same authors that swept U in perovskite vacancy calculations is curnan2014. See also xu2015c on linear response U for 3d oxides.