  In this work, the spectra of the prospective exotic hidden-charm and
hidden-bottom baryonium, viz. the baryon-antibaryon states, with
$J^{PC}=0^{--}$ and $0^{+-}$ are investigated in the framework of QCD sum
rules. The non-perturbative contributions up to dimension 12 are taken into
account. Numerical results indicate that there might exist 3 possible $0^{--}$
hidden-charm baryonium states with masses $(5.22\pm0.26)$, $(5.52\pm0.25)$, and
$(5.46\pm0.24)$ GeV, and 5 possible $0^{+-}$ hidden-charm baryonium states with
masses $(4.76\pm0.28)$, $(5.24\pm0.28)$, $(5.16\pm0.27)$, $(5.52\pm0.27)$, and
$(5.69\pm0.27)$ GeV, respectively. The corresponding hidden-bottom partners are
found lying in the range of $11.68-12.28$ GeV and $11.38-12.33$ GeV,
respectively. The possible baryonium decay modes are analyzed, which are
hopefully measurable in LHC experiments.
