We theoretically investigate long-range interactions between an excited L -state He atom and two identical S -state He atoms for the cases of the three atoms all in spin-singlet states or all in spin-triplet states, denoted by He(n0lambdaS )-He(n0lambdaS )-He(n0'lambdaL ), with n0 and n0' principal quantum numbers, lambda =1 or 3 the spin multiplicity, and L the orbital angular momentum of a He atom. Using degenerate perturbation theory for the energies up to second-order, we evaluate the coefficients C3 of the first-order dipolar interactions and the coefficients C6 and C8 of the second-order additive and nonadditive interactions. Both the dipolar and dispersion interaction coefficients, for these three-body degenerate systems, show dependences on the geometrical configurations of the three atoms. The nonadditive interactions start to appear in second-order. To demonstrate the results and for applications, the obtained coefficients Cn are evaluated with highly accurate variationally generated nonrelativistic wave functions in Hylleraas coordinates for He(1 1S ) -He(1 1S ) -He(2 1S ) , He(1 1S ) -He(1 1S ) -He(2 1P ) , He(2 1S ) -He(2 1S ) -He(2 1P ) , and He(2 3S ) -He(2 3S ) -He(2 3P ) . The calculations are given for three like nuclei for the cases of hypothetical infinite mass He nuclei, and of real finite mass 4He or 3He nuclei. The special cases of the three atoms in equilateral triangle configurations are explored in detail, and for the cases in which one of the atoms is in a P state, we also present results for the atoms in an isosceles right triangle configuration or in an equally spaced collinear configuration. The results can be applied to construct potential energy surfaces for three helium atom systems.
