Simple Impurity Embedded in a Spherical Jellium: Approximations of Density Functional Theory compared to Quantum Monte Carlo Benchmarks

TL;DR

This study compares various density functional theory approximations to quantum Monte Carlo benchmarks for a spherical jellium with a central impurity, revealing the limitations of DFT in capturing inhomogeneity effects and electronic transitions.

Contribution

It provides a systematic comparison of DFT functionals against QMC benchmarks for impurity-embedded spherical jellium, highlighting their strengths and weaknesses in different regimes.

Findings

Meta-GGA and hybrid functionals closely match QMC for 1d→2s transitions.

Larger discrepancies observed for 1f→2p transitions, especially in open-shell systems.

Increased inhomogeneity amplifies errors in DFT approximations.

Abstract

We study the electronic structure of a spherical jellium in the presence of a central Gaussian impurity. We test how well the resulting inhomogeneity effects beyond spherical jellium are reproduced by several approximations of density functional theory (DFT). Four rungs of Perdew's ladder of DFT functionals, namely local density approximation (LDA), generalized gradient approximation (GGA), meta-GGA and orbital-dependent hybrid functionals are compared against our quantum Monte Carlo (QMC) benchmarks. We identify several distinct transitions in the ground state of the system as the electronic occupation changes between delocalized and localized states. We examine the parameter space of realistic densities ($1 \le r_s\le 5$) and moderate depths of the Gaussian impurity ($Z<7$). The selected 18 electron system (with closed-shell ground state) presents $1d \to 2s$ transitions while the 30 electron system (with open-shell ground state) exhibits $1f \to 2p$ transitions. For the former system, the accuracy for the transitions is clearly improving with increasing sophistication of functionals with meta-GGA and hybrid functionals having only small deviations from QMC. However, for the latter system, we find much larger differences for the underlying transitions between our pool of DFT functionals and QMC. We attribute this failure to treatment of the exact exchange within these functionals. Additionally, we amplify the inhomogeneity effects by creating the system with spherical shell which leads to even larger errors in DFT approximations.