Explicitly correlated Gaussian (ECG) functions with carefully optimized non-linear parameters are used to calculate the electronic energies of He-2(+) and LiH at their equilibrium internuclear distances. The obtained variational upper bounds (-4.99464392 and -8.070538 hartree, respectively) are the lowest reported to date. By extrapolating results obtained with various expansion lengths, the estimations of the Born-Oppenheimer limits are made.