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

摘要

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 threeatoms all in spin singlet states or all in spin triplet states, denoted byHe($n_0\,^{\lambda}S$)-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 spinmultiplicity, and $L$ the orbital angular momentum of a He atom. Usingdegenerate perturbation theory for the energies up to second-order, we evaluatethe coefficients $C_3$ of the first order dipolar interactions and thecoefficients $C_6$ and $C_8$ of the second order additive and nonadditiveinteractions. Both the dipolar and dispersion interaction coefficients, forthese three-body degenerate systems, show dependences on the geometricalconfigurations of the three atoms. The nonadditive interactions start to appearin second-order. To demonstrate the results and for applications, the obtainedcoefficients $C_n$ are evaluated with highly accurate variationally-generatednonrelativistic wave functions in Hylleraas coordinates forHe($1\,^{1}S$)-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${(2\,^{1}P)}$, andHe${(2\,^{3}S)}$-He${(2\,^{3}S)}$-He${(2\,^{3}P)}$. The calculations are givenfor 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 thethree atoms in equilateral triangle configurations are explored in detail, andfor the cases where one of the atoms is in a $P$ state, we also present resultsfor the atoms in an isosceles right triangle configuration or in an equallyspaced co-linear configuration. The results can be applied to constructpotential energy surfaces for three helium atom systems.