General formulas for calculating the several leading long-range interactions among three identical atoms where two atoms are in identical S states and the other atom is in a P state are obtained using perturbation theory for the energies up to second order. The first order (dipolar) interactions depend on the geometrical configurations of the three atoms. In second order, additive and nonadditive dispersion interactions are obtained. The nonadditive interactions depend on the geometrical configurations in marked contrast to the case where all three atoms are in identical S states, for which the nonadditive (also known as triple-dipole or as Axilrod-Muto-Teller) dispersion interactions appear at the third order. The formalism is demonstrated by the calculation of the coefficients for the Li(2 2 S)-Li(2 2 S)-Li(2 2 P) system using variationally-generated atomic lithium wave functions in Hylleraas coordinates. The present dipolar coefficients and additive and nonadditive dispersion coefficients may be useful in constructing precise potential energy surfaces for this three lithium atom system.

The long-range interactions among two-or three-atom systems are of considerable importance in the cold and ultracold research areas for many body systems. For an ion and an atom, the longrange interaction potential is dominated by the induction (or polarization) potential resulting from the (classical) effect of the ion's electric field on the atom and the leading term of the induction potential is much stronger than the (quantum mechanical) dispersion (or van der Waals) interaction. The present paper focuses on the long-range interaction of the Li(2 2 S)-Li(2 2 S)-Li + (1 1 S) system, to see what changes this induction effect (originating in the electric field of the Li + ion) yields in the long-range additive and nonadditive interactions of this three-body system. Using perturbation theory for energies, we evaluate the coefficients C n in the potential energy for the three well-separated constituents, where n refers to the corresponding order in inverse powers of distance, obtaining the additive interaction coefficients C 4 , C 6 , C 7 , C 8 , C 9 and the nonadditive interaction coefficients C 7 , C 9 . The obtained coefficients C n are calculated with highly accurate variationally-generated nonrelativistic wave functions in Hylleraas coordinates. Our calculations may be of interest for the study of three-body recombination and for constructing precise potential energy surfaces. We also provide precise evaluations of the long-range potentials for the two-body Li(2 2 S)-Li + (1 1 S) system. For both the two-body and three-body cases, we provide results for the like-nuclei cases of 6 Li and 7 Li.