  The multi-component density functional theory is faced with the challenge of
capturing various types of inter- and intra-particle exchange-correlation
effects beyond those introduced by the conventional electronic
exchange-correlation functionals. Herein, we focus on evaluating the
electron-proton/muon correlation functionals appearing in
molecular/condensed-phase systems where a proton/muon is treated as a quantum
particle on equal footing with electrons, beyond the Born-Oppenheimer paradigm.
Five recently developed local correlation functionals, i.e., the epc series and
e$\mu$c-1, are selected and their performances are analyzed by employing a
two-particle model that includes an electron and a positively charged particle
(PCP) with a variable mass, interacting through Coulombic forces, within a
double harmonic trap. Using the Kohn-Sham (KS) inversion procedure, the exact
two-component KS characterization of the model is deduced and its properties
are compared to those derived from the considered functionals. The analysis
demonstrates that these local functionals achieve their original
parameterization objectives to reproduce the one-PCP densities and the
electron-PCP correlation energies, but all fall short of reproducing the
underlying PCP correlation potentials correctly. Moreover, a comprehensive
error analysis reveals that the density-driven errors have a non-negligible
contribution to the success of the considered functionals. Overall, the study
shows the strengths as well as shortcomings of the considered functionals
hopefully paving the way for designing more robust functionals in the future.
