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

TL;DR

This study reveals that DFT+$U$ simulations using PAW significantly underestimate high-pressure effects in lanthanide metals due to a normalization issue, highlighting limitations of current methods at ultra-high pressures.

Contribution

The paper systematically analyzes the failure of DFT+$U$ in PAW at high pressures for lanthanides, identifying the normalization problem of local density matrices as the cause.

Findings

DFT+$U$ in PAW underestimates force at high pressure.

Normalization problem of local density matrix causes unphysical results.

Breakdown of DFT+$U$ applicability in VASP at ultra-high pressures.

Abstract

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