  We report a theoretical study on the long-range additive and nonadditive
potentials for a three-body hybrid atom-atom-ion system composed of one ground
$S$ state Li atom, one excited $P$ state Li atom and one ground $S$ state
Li$^+$ ion, Li($2\,^{2}S$)-Li($2\,^{2}P$)-Li$^+(1\,^{1}S$). The interaction
coefficients are evaluated with highly accurate wave functions calculated
variationally in Hylleraas coordinates. For this hybrid system the three-body
nonadditive collective interactions (appearing in second-order) induced by the
energy degeneracy and enhanced by the induction effect of the Li$^+$ ion
through the internal electric field can be strong and even stronger than the
two-body additive interactions at the same order. We find that for particular
geometries the two-body additive interactions of the system sum to zero leaving
only three-body nonadditive collective interactions making the present system
potentially a platform to explore quantum three-body collective effects. We
also extract first-principles leading coefficients of the long-range
electrostatic, induction, and dispersion energies of Li$^+_2$ electronic states
correlating to Li($2\,^{2}P$)-Li$^+(1\,^{1}S$), which until now were not
available in the literature. The results should be especially valuable for the
exploration of schemes to create trimers with ultracold atoms and ions in
optical lattices.
