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Density functional theory plus $U$ (DFT+$U$) is one of the most efficientfirst-principles methods to simulate the cold pressure properties ofstrongly-correlated materials. However, the applicability of DFT+$U$ atultra-high pressure is not sufficiently studied, especially in the widely-usedaugmented schemes [such as projector augmented wave (PAW) and linearizedaugmented plane wave (LAPW)]. This work has systematically investigated theperformance of DFT+$U$ in PAW and LAPW at the pressure up to several hundredGPa for the lanthanide metals, which is a typical strongly-correlated series.We found DFT+$U$ simulation in PAW exhibits an unphysical underestimating offorce at high pressure. By delicate analysis and comparison withlocal-orbital-independent hybrid functional results, we have demonstrated thatthis unphysical behavior is related to a normalization problem on the localdensity matrix caused by the overlap of local orbitals in PAW under highpressure. Additionally, we observed a slight softening of force in DFT+$U$ inheavy lanthanides (Tm, Yb and Lu) at high pressure comparing with the DFTresults without the influence of local orbital overlap, and it might be relatedto the enhancement of bonding effect in correlation correction methods at highpressure. Our work reveals the underestimating of high pressure in DFT+$U$simulation, analyses two sources of this unusual behavior and proposes theirmechanism. Most importantly, our investigation highlights the breakdown ofDFT+$U$ for high pressure simulation in VASP package based on PAW framework.

The underestimation of high pressure in DFT+$U$ simulation for the wide range cold-pressure of lanthanide metals