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 $n_0^{\prime }$ 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 $\text{He}\left(1{}^{1}S\right)\text{−}\text{He}\left(1{}^{1}S\right)\text{−}\text{He}\left(2{}^{1}S\right)$, $\text{He}\left(1{}^{1}S\right)\text{−}\text{He}\left(1{}^{1}S\right)\text{−}\text{He}\left(2{}^{1}P\right)$, $\text{He}\left(2{}^{1}S\right)\text{−}\text{He}\left(2{}^{1}S\right)\text{−}\text{He}\left(2{}^{1}P\right)$, and $\text{He}\left(2{}^{3}S\right)\text{−}\text{He}\left(2{}^{3}S\right)\text{−}\text{He}\left(2{}^{3}P\right)$. The calculations are given for three like nuclei for the cases of hypothetical infinite mass He nuclei, and of real finite mass ${}^4\mathrm{He}$ or ${}^3\mathrm{He}$ 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.

Figure 1

Long-range potentials for the $\text{He}\left(1{}^{1}S\right)\text{−}\text{He}\left(1{}^{1}S\right)\text{−}\text{He}\left(2{}^{1}S\right)$ system for three different types of the zeroth-order wave functions, where the three atoms form an equilateral triangle, in atomic units. For each curve labeled by a wave function, the plotted curve is the sum of $\mathrm{\Delta }{E}^{\left(1\right)}$ and $\mathrm{\Delta }{E}^{\left(2\right)}$, where $\mathrm{\Delta }{E}^{\left(1\right)}=0$.

Figure 2

Long-range potentials for the $\text{He}\left(1{}^{1}S\right)\text{−}\text{He}\left(1{}^{1}S\right)\text{−}\text{He}\left(2{}^{1}P\right)$ system for three different types of the zeroth-order wave functions, where the three atoms form an equilateral triangle, in atomic units. For each curve labeled by a wave function, the plotted curve is the sum of $\mathrm{\Delta }{E}^{\left(1\right)}$ and $\mathrm{\Delta }{E}^{\left(2\right)}$.

Figure 3

Long-range potentials for the $\text{He}\left(2{}^{1}S\right)\text{−}\text{He}\left(2{}^{1}S\right)\text{−}\text{He}\left(2{}^{1}P\right)$ system for three different types of the zeroth-order wave functions, where the three atoms form an equilateral triangle, in atomic units. For each curve labeled by a wave function, the plotted curve is the sum of $\mathrm{\Delta }{E}^{\left(1\right)}$ and $\mathrm{\Delta }{E}^{\left(2\right)}$.

Figure 4

Long-range potentials for the $\text{He}\left(2{}^{3}S\right)\text{−}\text{He}\left(2{}^{3}S\right)\text{−}\text{He}\left(2{}^{3}P\right)$ system for three different types of the zeroth-order wave functions, where the three atoms form an equilateral triangle, in atomic units. For each curve labeled by a wave function, the plotted curve is the sum of $\mathrm{\Delta }{E}^{\left(1\right)}$ and $\mathrm{\Delta }{E}^{\left(2\right)}$.

Figure 5

Long-range potentials for the $\text{He}\left(1{}^{1}S\right)\text{−}\text{He}\left(1{}^{1}S\right)\text{−}\text{He}\left(2{}^{1}P\right)$ system for three different types of the zeroth-order wave functions, where the three atoms form a straight line, in atomic units. For each curve labeled by a wave function, the plotted curve is the sum of $\mathrm{\Delta }{E}^{\left(1\right)}$ and $\mathrm{\Delta }{E}^{\left(2\right)}$.

Figure 6

Long-range potentials for the $\text{He}\left(2{}^{1}S\right)\text{−}\text{He}\left(2{}^{1}S\right)\text{−}\text{He}\left(2{}^{1}P\right)$ system for three different types of the zeroth-order wave functions, where the three atoms form a straight line, in atomic units. For each curve labeled by a wave function, the plotted curve is the sum of $\mathrm{\Delta }{E}^{\left(1\right)}$ and $\mathrm{\Delta }{E}^{\left(2\right)}$.

Figure 7

Long-range potentials for the $\text{He}\left(2{}^{3}S\right)\text{−}\text{He}\left(2{}^{3}S\right)\text{−}\text{He}\left(2{}^{3}P\right)$ system for three different types of the zeroth-order wave functions, where the three atoms form a straight line, in atomic units. For each curve labeled by a wave function, the plotted curve is the sum of $\mathrm{\Delta }{E}^{\left(1\right)}$ and $\mathrm{\Delta }{E}^{\left(2\right)}$.