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We study the electronic structure of a spherical jellium in the presence of acentral Gaussian impurity. We test how well the resulting inhomogeneity effectsbeyond spherical jellium are reproduced by several approximations of densityfunctional 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 againstour quantum Monte Carlo (QMC) benchmarks. We identify several distincttransitions in the ground state of the system as the electronic occupationchanges between delocalized and localized states. We examine the parameterspace of realistic densities ($1 \le r_s\le 5$) and moderate depths of theGaussian impurity ($Z<7$). The selected 18 electron system (with closed-shellground state) presents $1d \to 2s$ transitions while the 30 electron system(with open-shell ground state) exhibits $1f \to 2p$ transitions. For the formersystem, the accuracy for the transitions is clearly improving with increasingsophistication of functionals with meta-GGA and hybrid functionals having onlysmall deviations from QMC. However, for the latter system, we find much largerdifferences for the underlying transitions between our pool of DFT functionalsand QMC. We attribute this failure to treatment of the exact exchange withinthese functionals. Additionally, we amplify the inhomogeneity effects bycreating the system with spherical shell which leads to even larger errors inDFT approximations.

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