Gauge-including projector augmented-wave NMR chemical shift calculations with DFT+$U$

We adapt the DFT+$U$ method in the gauge-including projector augmented-wave NMR chemical shift calculations within the plane wave pseudopotential implementation. The nonlocal Hubbard correction potential has been reexamined in order to comply with the gauge-including projector augmented-wave transformation under an external uniform magnetic field. The resulting expression is suitable for chemical shift calculations using both norm-conserving and ultrasoft pseudopotentials in the proector augmented-wave scheme. The implementation is applied to the ${}^{17}\mathrm{O}$ solid-state NMR chemical shift calculations for transition-metal and rare-earth oxides, including ${\mathrm{TiO}}_2$, ZnO, ${\mathrm{Ti}}_2{\mathrm{O}}_3, {\mathrm{La}}_2{\mathrm{O}}_3$, and ${\mathrm{CeO}}_2$. A comparison between the DFT and DFT+$U$ NMR chemical shifts for the selected materials is presented.

Figure 1

Calculated ${}^{17}\mathrm{O}$ NMR chemical shieldings in selected transition-metal and rare-earth oxides compared with experimental chemical shifts [65, 66, 67, 68] and previous results (brown cross symbols for silicates [25] and black crosses for alkaline-earth oxides [62]). Present results are marked with the triangular symbols, colored in blue and red for PBE and PBE+$U$ calculations, respectively. The dotted and dashed lines are linear regressions explained in the context.

Figure 2

Comparison of PBE and PBE+$U$ total density of states for (a) ${\mathrm{La}}_2{\mathrm{O}}_3$ and (b) ${\mathrm{CeO}}_2$. The major atomic characteristics are indicated near the spikes.