  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.
