The accurate ground-state potential energy functions of the beryllium lithium dimer, BeLi, and its cation, BeLi + , have been determined from ab initio calculations using the multi-reference averaged coupled-pair functional (MR-ACPF) method in conjunction with the correlation-consistent core-valence basis sets up to septuple-zeta quality. The importance of the electron correlation beyond the MR-ACPF level of approximation, scalar relativistic, and adiabatic effects was discussed. For BeLi + , the similar calculation was performed using the single-reference coupled-cluster method, up to the CCSDTQ level of approximation. The vibration-rotation energy levels of the two species were predicted to near the "spectroscopic" accuracy. Strangely enough, the predicted vibrational term values do not compare well with the recent experimental data for either BeLi or BeLi + , with discrepancies reaching as much as 20-60 cm -1 for highly excited vibrational levels.