Pseudopotentials, an overlooked source and remedy of DFT errors

TL;DR

This work reveals that pseudopotentials introduce atomic-level errors that affect DFT accuracy in a manner not fully correctable by XC functionals alone. By employing an atomic-level adjusted, intentionally inconsistent pseudopotential–XC scheme, the authors demonstrate substantial improvements in bandgap predictions for 54 Cu-containing semiconductors, dramatically reducing mean relative error from 80% to 20% and eliminating erroneous metallic predictions. The approach, which leverages hybrid-pseudopotentials for Cu while using a PBE functional for solids, outperforms both HSE and GW in accuracy and maintains computational efficiency similar to standard KS-DFT. The study reframes pseudopotentials as pivotal players in electronic structure, arguing that reproducing all-electron results with the exact XC requires core–valence XC inconsistency and clarifies why traditional consistency may hide fundamental physics behind bandgap problems.

Abstract

First-principles calculations rely heavily on pseudopotentials, yet their impact on accuracy is hardly addressed. In this work, we show that most pseudopotentials to date introduce errors, which manifest themselves as errors of atomic energy levels, leading to a $de facto$ deviation from the Hohenberg-Kohn theorem. We consider the atomic-level adjusted pseudopotentials, whose interplay with exchange-correlation functional provides a pragmatic correction that balances accuracy and efficiency. We benchmark our theory with bandgap calculation for 54 semiconductors containing monovalent Cu. The results, compared to those from conventional studies, not only remove all erroneous metal predictions for 11 compounds, but also reduce the mean relative error from 80\% to 20\%. Overall accuracy even exceeds those of standard hybrid functionals and GW methods.

Figures (4)

• Figure 1: (Color online) Schematic illustration of the difference in the consistency with respect to reproducing the all-electron calculations of the approximate XC and exact XC functionals using pseudopotential-DFT. The $r_0$ denotes the cut-off radius of the pseudopotential. Curves C1 (e.g., Hartree-Fock) and C2 (e.g., LDA/GGA) represent approximate XC functionals that are suitable for describing localized and delocalized electrons, respectively. Reproducing their all-electron calculations requires maintaining the consistency of the pseudopotential-XC approximation, i.e., the same XC functional is used on both sides of $r_0$. Curve C3 is assumed to be the exact XC, showing a transition of the electronic behavior from localized to delocalized at $r_0$ as the orbital radius increases. In the spirit of pseudopotentials, this is highly probable. Using the C1-XC to construct the pseudopotential for core electrons at $r < r_0$ while using the C2-XC to describe outer-shell valence electrons at $r > r_0$ can largely reproduce the all-electron calculations of the XC represented by the curve C4. Unambiguously, such "inconsistent" scheme better captures the physical essence of the exact XC of the curve C3, giving higher accuracy, and is therefore the correct interpretation of the consistency. Curve C5 represents the generally accepted computational paradigm (e.g., hybrid functional), which emphasizes only the optimization of the outer-shell valence electron XC functional while maintaining the LDA/GGA pseudopotential. The dashed connection of curves C4 and C5 at $r_0$ indicates the occurrence of a jump in the localization/delocalization characteristic. For clarity, we have slightly shifted the turning point of curve C5 away from $r_0$.
• Figure 2: (Color online) A schematic representation of the bandap underestimation in monovalent-Cu compounds due to atomic-level errors of the pseudopotential. We compare between the PBE-pseudopotential (Left panel) and hybrid-pseudopotential (Right panel), by taking Cu$_2$S as an example. Due to the tetrahedral crystal field, the Cu-3$d$ orbitals split into a higher triplet and a lower doublet. According to our first-principles calculations, the bandgap $E_g$ can be roughly approximated as $E_g = \Delta_{sd} - \Delta_{cf}$, where $\Delta_{sd}$ is the atomic 4$s$-3$d$ splitting energy of the Cu and $\Delta_{cf}$ is the raised energy from the anionic crystal field. The significant atomic-level error of the PBE-pseudopotential causes a too small $\Delta_{sd}$. When $\Delta_{sd} < \Delta_{cf}$, a negative $E_g$ is even given on the solid side, resulting in erroneous metal predictions. On the contrary, the use of hybrid-pseudopotential largely corrects the atomic-level error, giving a close experimental $\Delta_{sd}$, thus systematically fixing the bandgap underestimation.
• Figure 3: Calculated bandgaps of 54 monovalent-Cu semiconductors with the same PBE functional, but respectively using PBE-pseudopotential (orange hollow-triangles) and hybrid-pseudopotential (blue solid-balls), along with the known experimental values (green solid-rectangles). See Table S1 of the Supporting Information for detailsSI. Right panel shows 8 compounds for which the bandgap problem is known to be also associated with the outer-shell valence electrons. More relevant optical gaps are employed here for three delafossite transparent conductive oxides, namely, CuAlO$_2$, CuGaO$_2$ and CuInO$_2$. In spite of a zero global gap, by convention, the Cu 4$s$ and 3$d$ energy difference at the $\Gamma$ point is defined as a negative gap.
• Figure 4: (Color online) Calculated versus experimental bandgaps for 54 monovalent-Cu semiconductors, with the dashed line indicating perfect agreement between the two. The blue solid-ball data are from our hybrid-pseudopotential calculations, and the red semi-hollow-triangle (HSE) and black hollow-diamond (GW) data are from previous studies. See Table S1 of the Supporting Information for more detailsSI.