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Calculations of long-range three-body interactions for Li($2\,^2S$)-Li($2\,^2S$)-Li($2\,^2P$)

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($2\,^2S$)-Li($2\,^2S$)-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.