The interaction of Li and Li+ with a B40 all-boron fullerene was theoretically investigated at the B3LYP, and Minnesota 2006 levels of theory. It was found that, unexpectedly, the interaction Li+ cation with the electron deficient B40 fullerene is stronger than the Li atom. It indicates that the B40 fullerene does not act as a conventional Lewis acid because of its highly correlated structure. Frontier molecular orbitals, partial density of states, and natural bond orbital analyses were used to discuss this unusual behavior. Our calculations indicate that this behavior makes the B40 fullerene more appropriate for application in the Li-ion batteries as anode material. The calculated cell voltage is about 530mV. Also, it was found that Hartree Fock (HF) exchange percentage of density functionals has a reverse effect on the adsorption energies of Li and Li+. This energy is increased and decreased, respectively, for Li+ and Li adsorptions by increasing %HF exchange. Finally, a potential energy surface for Li and Li+ penetration into B40 fullerene was predicted.