Minimizing propagated density errors of atomic core-electron for simultaneously accurate bandgaps and lattice constants in closed-shell Copper semiconductors

Technical Summary: Minimizing Propagated Density Errors in Copper Semiconductors

Problem Statement
Density Functional Theory (DFT) is the standard for electronic structure calculations, yet it suffers from systematic errors due to violations of mathematical constraints in approximate exchange-correlation functionals. While these errors are well-known to produce incorrect electron densities even when total energies are accurate, the specific mechanism by which atomic core-electron density errors propagate into solid-state calculations remains poorly understood. In particular, standard pseudopotential-DFT approaches often generate pseudopotentials using (semi-)local functionals (LDA/GGA) that inherently contain self-interaction errors. These errors lead to an overly delocalized core-electron density, which erroneously enhances the quantum screening of the nuclear charge. Consequently, this propagates into solid-state calculations, causing significant inaccuracies in the energy positions of semi-core $3d$ electrons, bandgaps, and lattice constants, especially in Copper (Cu)-containing semiconductors.

Methodology
The authors conducted a systematic study of over 50 Cu-containing simple closed-shell semiconductors to investigate the impact of Cu core-electron density on bandgaps and lattice constants. The study utilized the Quantum ESPRESSO package with norm-conserving pseudopotentials generated via three distinct approaches:

1. LDA and PBE: Standard local density approximation and generalized gradient approximation functionals.
2. Modified Hartree-Fock (mHF): A hybrid functional combining exact Hartree-Fock exchange with PBE correlation ($E^{mHF}_{xc} = E^{HF}_{x} + E^{PBE}_{c}$).

The core innovation lies in a "real-space partitioning" strategy where different functionals are applied to different regions of the atom. Specifically, the authors employed mHF pseudopotentials for the Cu core (encoding the core-electron behavior) while retaining standard LDA or PBE functionals for the valence electrons in the solid-state calculation. This approach, denoted as mHF@LDA or mHF@PBE, treats the highly localized core electrons (where self-interaction is severe) with exact exchange, while treating the delocalized valence electrons with functionals suited for slowly varying densities.

Key Results

• Bandgap Accuracy: Conventional methods (LDA@LDA and PBE@PBE) systematically underestimate bandgaps, often predicting erroneous metallic states (11 and 14 cases, respectively, out of 54 systems). In contrast, the mHF@LDA and mHF@PBE approaches yield positive bandgaps for all systems. The mean relative error (MRE) for bandgaps drops from ~80% in conventional methods to 29% (mHF@LDA) and 20% (mHF@PBE).
• Lattice Constant Accuracy: While mHF@PBE improves bandgaps, it severely overestimates lattice constants (MRE ~3.7%). The mHF@LDA combination achieves simultaneous high accuracy for both properties, with an MRE of only 0.9% for lattice constants and a maximum deviation of ~3%, outperforming both pure LDA and PBE approaches which show systematic underestimation and overestimation, respectively.
• Atomic-Level Mechanism: Analysis of the isolated Cu atom reveals that mHF pseudopotentials cause the $3d$ electrons to contract toward the nucleus (reducing orbital radius and standard deviation) and the $4s$ electrons to become more delocalized. This increases the $3d$-$4s$ energy splitting ($\Delta_{sd}$) from ~0.7 eV (LDA) to ~4.35 eV (mHF), approaching the experimental value of 5.04 eV.
• Propagation to Solids: The improved core density in the mHF pseudopotential reduces the erroneous quantum screening of the nuclear attraction. This corrects the $3d$ energy levels in the solid, preventing the metallic prediction seen in LDA. Furthermore, the redistribution of charge affects bonding; the mHF approach reduces the electron density in the bonding region between Cu and anions (e.g., S in Cu$_2$S), leading to weaker binding energies and longer bond lengths, which corrects the systematic underestimation of lattice constants found in LDA.

Significance and Claims
The paper claims that the primary source of error in Cu-containing semiconductors is not the valence functional itself, but the propagated error from the core-electron density encoded in the pseudopotential. By employing a modified Hartree-Fock pseudopotential for the core while using standard functionals for the valence, the authors demonstrate that it is possible to achieve simultaneous high accuracy in both bandgaps and lattice constants across a broad class of materials.

The authors position pseudopotentials not merely as mathematical tools to accelerate all-electron calculations, but as active means to correct standard DFT errors. They argue that the exact exchange-correlation functional form may need to be spatially dependent—resembling mHF in the core region to eliminate self-interaction and LDA in the valence region to describe delocalized electrons—rather than a single functional form fitting all regions. This work suggests that future DFT developments should consider the core-valence partitioning more holistically, potentially leading to hybrid approaches where mixing parameters are functions of local electron density and its derivatives.