Calculations of long-range three-body interactions for He($n_0\,^{\lambda}S$)-He($n_0\,^{\lambda}S$)-He($n_0^{\prime}\,^{\lambda}L$)

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($n_0\,^{\lambda}S$)-He($n_0\,^{\lambda}S$)-He($n_0^{\prime}\,^{\lambda}L$), with $n_0$ and $n_0'$ 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 $C_3$ of the first order dipolar interactions and the coefficients $C_6$ and $C_8$ 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 $C_n$ are evaluated with highly accurate variationally-generated nonrelativistic wave functions in Hylleraas coordinates for He($1\,^{1}S$)-He($1\,^{1}S$)-He($2\,^{1}S$), He${(1\,^{1}S)}$-He${(1\,^{1}S)}$-He${(2\,^{1}P)}$, He${(2\,^{1}S)}$-He${(2\,^{1}S)}$-He${(2\,^{1}P)}$, and He${(2\,^{3}S)}$-He${(2\,^{3}S)}$-He${(2\,^{3}P)}$. The calculations are given for three like-nuclei for the cases of hypothetical infinite mass He nuclei, and of real finite mass $^4{}$He or $^3{}$He nuclei. The special cases of the three atoms in equilateral triangle configurations are explored in detail, and for the cases where 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 co-linear configuration. The results can be applied to construct potential energy surfaces for three helium atom systems.