  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.
