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

With the rapid developments in ultracold atomic and molecular physics, accurate determinations of long-range interactions between two and three atoms are important in, for example, analyzing atomic photoassociation. Long-range interactions of two-body systems were extensively studied for $S$ and $P$ state atoms, however, for three-body systems studies are limited to $S$-state atoms. In this work, a general formula for calculating the long-range interactions among three like atoms is presented using perturbation theory up to second order, where two atoms are in identical $S$ states and the other atom is in a $P$ state. Unlike the case where the three atoms are in identical $S$ states, here the first order interaction coefficients already show a dependence on the geometrical configuration of the three atoms, and nonadditive terms start to appear at the second order in energy corrections. For the Li($2\,^2S$)-Li($2\,^2S$)-Li($2\,^{2}P$) system, we perform precision evaluation of various dispersion coefficients using variationally generated atomic lithium wave functions in Hylleraas coordinates. These additive and nonadditive long-range dispersion coefficients may be useful in constructing a precise potential energy surface of this three lithium system.